let R be Ring; :: thesis: for V being LeftMod of R
for l being Linear_Combination of V
for X being Subset of V st X misses Carrier l holds
l .: X c= {(0. R)}
let V be LeftMod of R; :: thesis: for l being Linear_Combination of V
for X being Subset of V st X misses Carrier l holds
l .: X c= {(0. R)}
let l be Linear_Combination of V; :: thesis: for X being Subset of V st X misses Carrier l holds
l .: X c= {(0. R)}
let X be Subset of V; :: thesis: ( X misses Carrier l implies l .: X c= {(0. R)} )
assume A1: X misses Carrier l ; :: thesis: l .: X c= {(0. R)}
set M = l .: X;
for y being object st y in l .: X holds
y in {(0. R)}

proof

let y be object ; :: thesis: ( y in l .: X implies y in {(0. R)} )
assume y in l .: X ; :: thesis: y in {(0. R)}
then consider x being object such that
x in dom l and
A2: x in X and
A3: y = l . x by FUNCT_1:def 6;
reconsider x = x as Element of V by A2;

now :: thesis: not l . x <> 0. R

assume l . x <> 0. R ; :: thesis: contradiction
then x in Carrier l ;
hence contradiction by A1, A2, XBOOLE_0:def 4; :: thesis: verum

end;

hence y in {(0. R)} by A3, TARSKI:def 1; :: thesis: verum

end;

hence l .: X c= {(0. R)} ; :: thesis: verum