let F be Field; :: thesis: for V being VectSp of F

for l being Linear_Combination of V

for X being Subset of V st X misses Carrier l holds

l .: X c= {(0. F)}

let V be VectSp of F; :: thesis: for l being Linear_Combination of V

for X being Subset of V st X misses Carrier l holds

l .: X c= {(0. F)}

let l be Linear_Combination of V; :: thesis: for X being Subset of V st X misses Carrier l holds

l .: X c= {(0. F)}

let X be Subset of V; :: thesis: ( X misses Carrier l implies l .: X c= {(0. F)} )

assume A1: X misses Carrier l ; :: thesis: l .: X c= {(0. F)}

set M = l .: X;

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in l .: X or y in {(0. F)} )

assume y in l .: X ; :: thesis: y in {(0. F)}

then consider x being object such that

A2: x in dom l and

A3: x in X and

A4: y = l . x by FUNCT_1:def 6;

reconsider x = x as Element of V by A2;

for l being Linear_Combination of V

for X being Subset of V st X misses Carrier l holds

l .: X c= {(0. F)}

let V be VectSp of F; :: thesis: for l being Linear_Combination of V

for X being Subset of V st X misses Carrier l holds

l .: X c= {(0. F)}

let l be Linear_Combination of V; :: thesis: for X being Subset of V st X misses Carrier l holds

l .: X c= {(0. F)}

let X be Subset of V; :: thesis: ( X misses Carrier l implies l .: X c= {(0. F)} )

assume A1: X misses Carrier l ; :: thesis: l .: X c= {(0. F)}

set M = l .: X;

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in l .: X or y in {(0. F)} )

assume y in l .: X ; :: thesis: y in {(0. F)}

then consider x being object such that

A2: x in dom l and

A3: x in X and

A4: y = l . x by FUNCT_1:def 6;

reconsider x = x as Element of V by A2;

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

hence
y in {(0. F)}
by A4, TARSKI:def 1; :: thesis: verumassume
l . x <> 0. F
; :: thesis: contradiction

then x in Carrier l ;

then x in (Carrier l) /\ X by A3, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

end;then x in Carrier l ;

then x in (Carrier l) /\ X by A3, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum