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 Th19;
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, Th22; :: thesis: verum