Etymology
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lean (n.1)
"action or state of leaning, deviation from a vertical position," 1776, from lean (v.).
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lean (n.2)
c. 1200, "lean animals or persons," from lean (adj.). Meaning "lean part of anything, muscle without fat, lean meat" is mid-15c.
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lean (v.)
c. 1200, from Old English hlinian "to recline, lie down, rest; bend or incline" (Mercian hleonian, Northumbrian hlionian), from Proto-Germanic *hlinen (source also of Old Saxon hlinon, Old Frisian lena, Middle Dutch lenen, Dutch leunen, Old High German hlinen, German lehnen "to lean"), from PIE root *klei- "to lean."

Transitive sense "cause to lean or rest" is from 14c. Meaning "to incline the body against something for support" is mid-13c. Figurative sense of "to trust for support" is from early 13c. Sense of "to lean toward mentally, to favor" is from late 14c. Related: Leaned; leaning. Colloquial lean on "put pressure on" (someone) is first recorded 1960.
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lean (adj.)
"thin, spare, with little flesh or fat," c. 1200, from Old English hlæne "lean, thin," possibly (Skeat) from hlænan "cause to lean or bend," from Proto-Germanic *khlainijan, which would connect it to Old English hleonian (see lean (v.)). But perhaps rather, according to OED, from a PIE *qloinio- (with cognates in Lithuanian klynas "scrap, fragment," Lettish kleins "feeble"). Extended and figurative senses from early 14c. In business jargon, paired with mean (adj.) from 1970s to suggest aggressiveness as if from hunger.
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lean-to (n.)
"building whose rafters lean against another building or wall," mid-15c., from lean (v.) + to (adv.). Compare penthouse. "An addition made to a house behind, or at the end of it, chiefly for domestic offices, of one story or more, lower than the main building, and the roof of it leaning against the wall of the house" [Bartlett].
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leant 
occasion past tense and past participle of lean (v.).
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leanness (n.)
Old English hlænnesse; see lean (adj.) + -ness.
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leaning (n.)
Old English hlininga; verbal noun from lean (v.). Figurative sense "inclination, tendency" is from 1580s. Related: Leanings.
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links (n.)
"undulating sandy ground," 1728, from Scottish/Northumbrian link "sandy, rolling ground near seashore, a crook or winding of a river," from Old English hlinc "rising ground, ridge;" perhaps from the same Proto-Germanic root as lean (v.). The Scottish word for the type of landscape where golf was born; the word has been part of the names of golf courses at least since 1728. The southern form of the word was Middle English linch "rising ground, especially between plowed fields or along a chalk down," which persisted in dialect.
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*klei- 

Proto-Indo-European root meaning "to lean."

It forms all or part of: acclivity; anticline; clemency; client; climate; climax; cline; clinic; clinical; clino-; clitellum; clitoris; decline; declivity; enclitic; heteroclite; incline; ladder; lean (v.); lid; low (n.2) "small hill, eminence;" matroclinous; patroclinous; polyclinic; proclitic; proclivity; recline; synclinal; thermocline.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit srayati "leans," sritah "leaning;" Old Persian cay "to lean;" Lithuanian šlyti "to slope," šlieti "to lean;" Latin clinare "to lean, bend," clivus "declivity," inclinare "cause to bend," declinare "bend down, turn aside;" Greek klinein "to cause to slope, slant, incline;" Old Irish cloin "crooked, wrong;" Middle Irish cle, Welsh cledd "left," literally "slanting").

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