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,REAL st
( ( for a, b being Real st (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) = RealFuncZero A holds
( a = 0 & b = 0 ) ) & ( for h being Element of Funcs A,REAL ex a, b being Real st h = (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) ) )
let A be non empty set ; :: thesis: ( A = {x1,x2} & x1 <> x2 implies ex f, g being Element of Funcs A,REAL st
( ( for a, b being Real st (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) = RealFuncZero A holds
( a = 0 & b = 0 ) ) & ( for h being Element of Funcs A,REAL ex a, b being Real st h = (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) ) ) )
assume A1: ( A = {x1,x2} & x1 <> x2 ) ; :: thesis: ex f, g being Element of Funcs A,REAL st
( ( for a, b being Real st (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) = RealFuncZero A holds
( a = 0 & b = 0 ) ) & ( for h being Element of Funcs A,REAL ex a, b being Real st h = (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) ) )
then A2: ( x1 in A & x2 in A & x1 <> x2 ) by TARSKI:def 2;
consider f, g being Element of Funcs A,REAL such that
A3: for z being set st z in A holds
( ( z = x1 implies f . z = 1 ) & ( z <> x1 implies f . z = 0 ) ) and
A4: for z being set st z in A holds
( ( z = x1 implies g . z = 0 ) & ( z <> x1 implies g . z = 1 ) ) by Th28;
take f ; :: thesis: ex g being Element of Funcs A,REAL st
( ( for a, b being Real st (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) = RealFuncZero A holds
( a = 0 & b = 0 ) ) & ( for h being Element of Funcs A,REAL ex a, b being Real st h = (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) ) )
take g ; :: thesis: ( ( for a, b being Real st (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) = RealFuncZero A holds
( a = 0 & b = 0 ) ) & ( for h being Element of Funcs A,REAL ex a, b being Real st h = (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) ) )
thus ( ( for a, b being Real st (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) = RealFuncZero A holds
( a = 0 & b = 0 ) ) & ( for h being Element of Funcs A,REAL ex a, b being Real st h = (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) ) ) by A1, A2, A3, A4, Th29, Th31; :: thesis: verum