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 = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [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 = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [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 = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [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 = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))

take g ; :: thesis: for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))

let h be Element of Funcs (A,COMPLEX); :: thesis: ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))

thus ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g])) by A1, A2, Th20; :: thesis: verum

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 = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [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 = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [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 = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [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 = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))

take g ; :: thesis: for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))

let h be Element of Funcs (A,COMPLEX); :: thesis: ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))

thus ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g])) by A1, A2, Th20; :: thesis: verum