let x1, x2 be set ; :: thesis: for A being non empty set
for f, g being Element of Funcs (A,COMPLEX) st A = {x1,x2} & 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 h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = () . (( . [a,f]),( . [b,g]))

let A be non empty set ; :: thesis: for f, g being Element of Funcs (A,COMPLEX) st A = {x1,x2} & 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 h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = () . (( . [a,f]),( . [b,g]))

let f, g be Element of Funcs (A,COMPLEX); :: thesis: ( A = {x1,x2} & 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 h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = () . (( . [a,f]),( . [b,g])) )

assume that
A1: A = {x1,x2} and
A2: x1 <> x2 and
A3: ( ( 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 h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = () . (( . [a,f]),( . [b,g]))
x2 in A by ;
then A4: ( f . x2 = 0c & g . x2 = 1r ) by A2, A3;
x1 in A by ;
then A5: ( f . x1 = 1r & g . x1 = 0c ) by A3;
let h be Element of Funcs (A,COMPLEX); :: thesis: ex a, b being Complex st h = () . (( . [a,f]),( . [b,g]))
reconsider x1 = x1, x2 = x2 as Element of A by ;
take a = h . x1; :: thesis: ex b being Complex st h = () . (( . [a,f]),( . [b,g]))
take b = h . x2; :: thesis: h = () . (( . [a,f]),( . [b,g]))
now :: thesis: for x being Element of A holds h . x = (() . (( . [a,f]),( . [b,g]))) . x
let x be Element of A; :: thesis: h . x = (() . (( . [a,f]),( . [b,g]))) . x
A6: ( x = x1 or x = x2 ) by ;
A7: (() . (( . [a,f]),( . [b,g]))) . x2 = (( . [a,f]) . x2) + (( . [b,g]) . x2) by Th1
.= (a * (f . x2)) + (( . [b,g]) . x2) by Th4
.= 0c + (b * 1r) by
.= h . x2 by COMPLEX1:def 4 ;
(() . (( . [a,f]),( . [b,g]))) . x1 = (( . [a,f]) . x1) + (( . [b,g]) . x1) by Th1
.= (a * (f . x1)) + (( . [b,g]) . x1) by Th4
.= a + (b * 0c) by
.= h . x1 ;
hence h . x = (() . (( . [a,f]),( . [b,g]))) . x by A6, A7; :: thesis: verum
end;
hence h = () . (( . [a,f]),( . [b,g])) by FUNCT_2:63; :: thesis: verum