let A be non empty set ; :: thesis: for f, g, h, f1 being Element of Funcs (A,REAL)
for x1, x2, x3, x4 being Element of A st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & 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 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) holds
for a, b, c, d being Real st () . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )

let f, g, h, f1 be Element of Funcs (A,REAL); :: thesis: for x1, x2, x3, x4 being Element of A st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & 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 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) holds
for a, b, c, d being Real st () . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )

let x1, x2, x3, x4 be Element of A; :: thesis: ( x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & 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 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) implies for a, b, c, d being Real st () . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 ) )

set RM = RealFuncExtMult A;
set RA = RealFuncAdd A;
assume that
A1: x1 <> x2 and
A2: x1 <> x3 and
A3: x1 <> x4 and
A4: x2 <> x3 and
A5: x2 <> x4 and
A6: x3 <> x4 and
A7: f . x1 = 1 and
A8: for z being set st z in A & z <> x1 holds
f . z = 0 and
A9: g . x2 = 1 and
A10: for z being set st z in A & z <> x2 holds
g . z = 0 and
A11: h . x3 = 1 and
A12: for z being set st z in A & z <> x3 holds
h . z = 0 and
A13: f1 . x4 = 1 and
A14: for z being set st z in A & z <> x4 holds
f1 . z = 0 ; :: thesis: for a, b, c, d being Real st () . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1])) = RealFuncZero A holds
( a = 0 & b = 0 & c = 0 & d = 0 )

A15: ( f . x2 = 0 & h . x2 = 0 ) by A1, A4, A8, A12;
A16: ( g . x1 = 0 & h . x1 = 0 ) by A1, A2, A10, A12;
A17: f1 . x1 = 0 by ;
A18: f1 . x2 = 0 by ;
let a, b, c, d be Real; :: thesis: ( () . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1])) = RealFuncZero A implies ( a = 0 & b = 0 & c = 0 & d = 0 ) )
assume A19: (RealFuncAdd A) . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1])) = RealFuncZero A ; :: thesis: ( a = 0 & b = 0 & c = 0 & d = 0 )
reconsider a = a, b = b, c = c, d = d as Element of REAL by XREAL_0:def 1;
A20: 0 = (() . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1]))) . x2 by
.= ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x2) + ((() . [d,f1]) . x2) by FUNCSDOM:1
.= (((() . ((() . [a,f]),(() . [b,g]))) . x2) + ((() . [c,h]) . x2)) + ((() . [d,f1]) . x2) by FUNCSDOM:1
.= ((((() . [a,f]) . x2) + ((() . [b,g]) . x2)) + ((() . [c,h]) . x2)) + ((() . [d,f1]) . x2) by FUNCSDOM:1
.= ((((() . [a,f]) . x2) + ((() . [b,g]) . x2)) + ((() . [c,h]) . x2)) + (d * (f1 . x2)) by FUNCSDOM:4
.= ((((() . [a,f]) . x2) + ((() . [b,g]) . x2)) + (c * (h . x2))) + (d * (f1 . x2)) by FUNCSDOM:4
.= ((((() . [a,f]) . x2) + (b * (g . x2))) + (c * (h . x2))) + (d * (f1 . x2)) by FUNCSDOM:4
.= (((a * 0) + (b * 1)) + (c * 0)) + (d * 0) by
.= b ;
A21: ( f . x4 = 0 & g . x4 = 0 ) by A3, A5, A8, A10;
A22: h . x4 = 0 by ;
A23: ( f . x3 = 0 & g . x3 = 0 ) by A2, A4, A8, A10;
A24: f1 . x3 = 0 by ;
A25: 0 = (() . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1]))) . x4 by
.= ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x4) + ((() . [d,f1]) . x4) by FUNCSDOM:1
.= (((() . ((() . [a,f]),(() . [b,g]))) . x4) + ((() . [c,h]) . x4)) + ((() . [d,f1]) . x4) by FUNCSDOM:1
.= ((((() . [a,f]) . x4) + ((() . [b,g]) . x4)) + ((() . [c,h]) . x4)) + ((() . [d,f1]) . x4) by FUNCSDOM:1
.= ((((() . [a,f]) . x4) + ((() . [b,g]) . x4)) + ((() . [c,h]) . x4)) + (d * (f1 . x4)) by FUNCSDOM:4
.= ((((() . [a,f]) . x4) + ((() . [b,g]) . x4)) + (c * (h . x4))) + (d * (f1 . x4)) by FUNCSDOM:4
.= ((((() . [a,f]) . x4) + (b * (g . x4))) + (c * (h . x4))) + (d * (f1 . x4)) by FUNCSDOM:4
.= (((a * 0) + (b * 0)) + (c * 0)) + (d * 1) by
.= d ;
A26: 0 = (() . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1]))) . x3 by
.= ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x3) + ((() . [d,f1]) . x3) by FUNCSDOM:1
.= (((() . ((() . [a,f]),(() . [b,g]))) . x3) + ((() . [c,h]) . x3)) + ((() . [d,f1]) . x3) by FUNCSDOM:1
.= ((((() . [a,f]) . x3) + ((() . [b,g]) . x3)) + ((() . [c,h]) . x3)) + ((() . [d,f1]) . x3) by FUNCSDOM:1
.= ((((() . [a,f]) . x3) + ((() . [b,g]) . x3)) + ((() . [c,h]) . x3)) + (d * (f1 . x3)) by FUNCSDOM:4
.= ((((() . [a,f]) . x3) + ((() . [b,g]) . x3)) + (c * (h . x3))) + (d * (f1 . x3)) by FUNCSDOM:4
.= ((((() . [a,f]) . x3) + (b * (g . x3))) + (c * (h . x3))) + (d * (f1 . x3)) by FUNCSDOM:4
.= (((a * 0) + (b * 0)) + (c * 1)) + (d * 0) by
.= c ;
0 = (() . ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))),(() . [d,f1]))) . x1 by
.= ((() . ((() . ((() . [a,f]),(() . [b,g]))),(() . [c,h]))) . x1) + ((() . [d,f1]) . x1) by FUNCSDOM:1
.= (((() . ((() . [a,f]),(() . [b,g]))) . x1) + ((() . [c,h]) . x1)) + ((() . [d,f1]) . x1) by FUNCSDOM:1
.= ((((() . [a,f]) . x1) + ((() . [b,g]) . x1)) + ((() . [c,h]) . x1)) + ((() . [d,f1]) . x1) by FUNCSDOM:1
.= ((((() . [a,f]) . x1) + ((() . [b,g]) . x1)) + ((() . [c,h]) . x1)) + (d * (f1 . x1)) by FUNCSDOM:4
.= ((((() . [a,f]) . x1) + ((() . [b,g]) . x1)) + (c * (h . x1))) + (d * (f1 . x1)) by FUNCSDOM:4
.= ((((() . [a,f]) . x1) + (b * (g . x1))) + (c * (h . x1))) + (d * (f1 . x1)) by FUNCSDOM:4
.= (((a * 1) + (b * 0)) + (c * 0)) + (d * 0) by
.= a ;
hence ( a = 0 & b = 0 & c = 0 & d = 0 ) by ; :: thesis: verum