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

assume A1: ( A = {x1,x2} & x1 <> x2 ) ; :: thesis: ex f, g being Element of Funcs (A,COMPLEX) st
for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = () . (( . [a,f]),( . [b,g]))

consider f, g being Element of Funcs (A,COMPLEX) such that
A2: ( ( for z being set st z in A holds
( ( z = x1 implies f . z = 1r ) & ( z <> x1 implies f . z = 0c ) ) ) & ( for z being set st z in A holds
( ( z = x1 implies g . z = 0c ) & ( z <> x1 implies g . z = 1r ) ) ) ) by Th17;
take f ; :: thesis: ex g being Element of Funcs (A,COMPLEX) st
for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = () . (( . [a,f]),( . [b,g]))

take g ; :: thesis: for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = () . (( . [a,f]),( . [b,g]))
let h be Element of Funcs (A,COMPLEX); :: thesis: ex a, b being Complex st h = () . (( . [a,f]),( . [b,g]))
thus ex a, b being Complex st h = () . (( . [a,f]),( . [b,g])) by A1, A2, Th20; :: thesis: verum