let x1, x2 be set ; :: thesis: for A being non empty set
for f, g being Element of Funcs (A,COMPLEX) st x1 in A & x2 in A & x1 <> x2 & ( for z being set st z in A holds
( ( z = x1 implies f . z = 1r ) & ( z <> x1 implies f . z = 0 ) ) ) & ( for z being set st z in A holds
( ( z = x1 implies g . z = 0 ) & ( z <> x1 implies g . z = 1r ) ) ) holds
for a, b being Complex st () . (( . [a,f]),( . [b,g])) = ComplexFuncZero A holds
( a = 0c & b = 0c )

let A be non empty set ; :: thesis: for f, g being Element of Funcs (A,COMPLEX) st x1 in A & x2 in A & x1 <> x2 & ( for z being set st z in A holds
( ( z = x1 implies f . z = 1r ) & ( z <> x1 implies f . z = 0 ) ) ) & ( for z being set st z in A holds
( ( z = x1 implies g . z = 0 ) & ( z <> x1 implies g . z = 1r ) ) ) holds
for a, b being Complex st () . (( . [a,f]),( . [b,g])) = ComplexFuncZero A holds
( a = 0c & b = 0c )

let f, g be Element of Funcs (A,COMPLEX); :: thesis: ( x1 in A & x2 in A & x1 <> x2 & ( for z being set st z in A holds
( ( z = x1 implies f . z = 1r ) & ( z <> x1 implies f . z = 0 ) ) ) & ( for z being set st z in A holds
( ( z = x1 implies g . z = 0 ) & ( z <> x1 implies g . z = 1r ) ) ) implies for a, b being Complex st () . (( . [a,f]),( . [b,g])) = ComplexFuncZero A holds
( a = 0c & b = 0c ) )

assume that
A1: x1 in A and
A2: x2 in A and
A3: x1 <> x2 and
A4: ( ( for z being set st z in A holds
( ( z = x1 implies f . z = 1r ) & ( z <> x1 implies f . z = 0 ) ) ) & ( for z being set st z in A holds
( ( z = x1 implies g . z = 0 ) & ( z <> x1 implies g . z = 1r ) ) ) ) ; :: thesis: for a, b being Complex st () . (( . [a,f]),( . [b,g])) = ComplexFuncZero A holds
( a = 0c & b = 0c )

A5: ( f . x2 = 0c & g . x2 = 1r ) by A2, A3, A4;
A6: ( f . x1 = 1r & g . x1 = 0c ) by A1, A4;
let a, b be Complex; :: thesis: ( () . (( . [a,f]),( . [b,g])) = ComplexFuncZero A implies ( a = 0c & b = 0c ) )
reconsider x1 = x1, x2 = x2 as Element of A by A1, A2;
assume A7: (ComplexFuncAdd A) . (( . [a,f]),( . [b,g])) = ComplexFuncZero A ; :: thesis: ( a = 0c & b = 0c )
reconsider a = a, b = b as Element of COMPLEX by XCMPLX_0:def 2;
A8: 0c = (() . (( . [a,f]),( . [b,g]))) . x2 by
.= (( . [a,f]) . x2) + (( . [b,g]) . x2) by Th1
.= (a * (f . x2)) + (( . [b,g]) . x2) by Th4
.= 0c + (b * 1r) by
.= b by COMPLEX1:def 4 ;
0c = (() . (( . [a,f]),( . [b,g]))) . x1 by
.= (( . [a,f]) . x1) + (( . [b,g]) . x1) by Th1
.= (a * (f . x1)) + (( . [b,g]) . x1) by Th4
.= a + (b * 0c) by
.= a ;
hence ( a = 0c & b = 0c ) by A8; :: thesis: verum