let R be Ring; :: thesis: for V being LeftMod of R
for l being Linear_Combination of V
for A being Subset of V
for v being Element of V st v in A holds
(l ! A) . v = l . v
let V be LeftMod of R; :: thesis: for l being Linear_Combination of V
for A being Subset of V
for v being Element of V st v in A holds
(l ! A) . v = l . v
let l be Linear_Combination of V; :: thesis: for A being Subset of V
for v being Element of V st v in A holds
(l ! A) . v = l . v
let A be Subset of V; :: thesis: for v being Element of V st v in A holds
(l ! A) . v = l . v
let v be Element of V; :: thesis: ( v in A implies (l ! A) . v = l . v )
assume A1: v in A ; :: thesis: (l ! A) . v = l . v
dom (l ! A) = [#] V by FUNCT_2:92;
then A2: (dom (l | A)) \/ (dom ((A `) --> (0. R))) = [#] V by FUNCT_4:def 1;
not v in dom ((A `) --> (0. R)) by A1, XBOOLE_0:def 5;
then (l ! A) . v = (l | A) . v by A2, FUNCT_4:def 1
.= l . v by A1, FUNCT_1:49 ;
hence (l ! A) . v = l . v ; :: thesis: verum