let A be non empty set ; :: thesis: for f, g, h being Element of Funcs A,REAL
for x1, x2, x3 being Element of A st x1 in A & x2 in A & x3 in A & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for a, b, c being Real st (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h]) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
let f, g, h be Element of Funcs A,REAL ; :: thesis: for x1, x2, x3 being Element of A st x1 in A & x2 in A & x3 in A & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) holds
for a, b, c being Real st (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h]) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
let x1, x2, x3 be Element of A; :: thesis: ( x1 in A & x2 in A & x3 in A & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) implies for a, b, c being Real st (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h]) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 ) )
set RM = RealFuncExtMult A;
set RA = RealFuncAdd A;
assume that
A1: ( x1 in A & x2 in A & x3 in A & x1 <> x2 & x1 <> x3 & x2 <> x3 ) and
A2: ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds
f . z = 0 ) ) and
A3: ( g . x2 = 1 & ( for z being set st z in A & z <> x2 holds
g . z = 0 ) ) and
A4: ( h . x3 = 1 & ( for z being set st z in A & z <> x3 holds
h . z = 0 ) ) ; :: thesis: for a, b, c being Real st (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h]) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 )
A5: ( f . x1 = 1 & f . x2 = 0 & f . x3 = 0 & g . x1 = 0 & g . x2 = 1 & g . x3 = 0 & h . x1 = 0 & h . x2 = 0 & h . x3 = 1 ) by A1, A2, A3, A4;
let a, b, c be Real; :: thesis: ( (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h]) = RealFuncZero A implies ( a = 0 & b = 0 & c = 0 ) )
assume A6: (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h]) = RealFuncZero A ; :: thesis: ( a = 0 & b = 0 & c = 0 )
then A7: 0 = ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])) . x1 by FUNCOP_1:13
.= (((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) . x1) + (((RealFuncExtMult A) . [c,h]) . x1) by FUNCSDOM:10
.= ((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1) by FUNCSDOM:10
.= ((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (c * (h . x1)) by FUNCSDOM:15
.= ((((RealFuncExtMult A) . [a,f]) . x1) + (b * (g . x1))) + (c * (h . x1)) by FUNCSDOM:15
.= ((a * 1) + (b * 0 )) + (c * 0 ) by A5, FUNCSDOM:15
.= a ;
A8: 0 = ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])) . x2 by A6, FUNCOP_1:13
.= (((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) . x2) + (((RealFuncExtMult A) . [c,h]) . x2) by FUNCSDOM:10
.= ((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2) by FUNCSDOM:10
.= ((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (c * (h . x2)) by FUNCSDOM:15
.= ((((RealFuncExtMult A) . [a,f]) . x2) + (b * (g . x2))) + (c * (h . x2)) by FUNCSDOM:15
.= ((a * 0 ) + (b * 1)) + (c * 0 ) by A5, FUNCSDOM:15
.= b ;
0 = ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])) . x3 by A6, FUNCOP_1:13
.= (((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) . x3) + (((RealFuncExtMult A) . [c,h]) . x3) by FUNCSDOM:10
.= ((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3) by FUNCSDOM:10
.= ((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (c * (h . x3)) by FUNCSDOM:15
.= ((((RealFuncExtMult A) . [a,f]) . x3) + (b * (g . x3))) + (c * (h . x3)) by FUNCSDOM:15
.= ((a * 0 ) + (b * 0 )) + (c * 1) by A5, FUNCSDOM:15
.= c ;
hence ( a = 0 & b = 0 & c = 0 ) by A7, A8; :: thesis: verum