let x1, x2 be set ; :: thesis: for A being non empty set st x1 in A & x2 in A & 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 )
let A be non empty set ; :: thesis: ( x1 in A & x2 in A & 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 ) )
assume A1: ( x1 in A & x2 in A & 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 )
consider f, g being Element of Funcs A,REAL such that
A2: ( ( for z being set st z in A holds
( ( z = x1 implies f . z = 1 ) & ( 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 = 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 )
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 )
let a, b be Real; :: thesis: ( (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) = RealFuncZero A implies ( a = 0 & b = 0 ) )
assume (RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]) = RealFuncZero A ; :: thesis: ( a = 0 & b = 0 )
hence ( a = 0 & b = 0 ) by A1, A2, Th29; :: thesis: verum