consider A being non empty set , x1, x2 being set such that
A1: ( A = {x1,x2} & x1 <> x2 ) by Lm3;
take V = ComplexVectSpace A; :: thesis: ex u, v being VECTOR of V st
( ( for a, b being Complex st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) & ( for w being VECTOR of V ex a, b being Complex st w = (a * u) + (b * v) ) )

consider f, g being Element of Funcs (A,COMPLEX) such that
A2: for a, b being Complex st () . (( . [a,f]),( . [b,g])) = ComplexFuncZero A holds
( a = 0c & b = 0c ) and
A3: for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = () . (( . [a,f]),( . [b,g])) by ;
reconsider u = f, v = g as VECTOR of V ;
take u ; :: thesis: ex v being VECTOR of V st
( ( for a, b being Complex st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) & ( for w being VECTOR of V ex a, b being Complex st w = (a * u) + (b * v) ) )

take v ; :: thesis: ( ( for a, b being Complex st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) ) & ( for w being VECTOR of V ex a, b being Complex st w = (a * u) + (b * v) ) )

thus for a, b being Complex st (a * u) + (b * v) = 0. V holds
( a = 0 & b = 0 ) by A2; :: thesis: for w being VECTOR of V ex a, b being Complex st w = (a * u) + (b * v)
thus for w being VECTOR of V ex a, b being Complex st w = (a * u) + (b * v) :: thesis: verum
proof
let w be VECTOR of V; :: thesis: ex a, b being Complex st w = (a * u) + (b * v)
reconsider h = w as Element of Funcs (A,COMPLEX) ;
consider a, b being Complex such that
A4: h = () . (( . [a,f]),( . [b,g])) by A3;
h = (a * u) + (b * v) by A4;
hence ex a, b being Complex st w = (a * u) + (b * v) ; :: thesis: verum
end;