let A be non empty set ; for f, g, h, f1 being Element of Funcs (A,REAL)
for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & 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 h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
let f, g, h, f1 be Element of Funcs (A,REAL); for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & 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 h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
let x1, x2, x3, x4 be Element of A; ( A = {x1,x2,x3,x4} & 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 h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) )
assume that
A1:
A = {x1,x2,x3,x4}
and
A2:
x1 <> x2
and
A3:
x1 <> x3
and
A4:
x1 <> x4
and
A5:
x2 <> x3
and
A6:
x2 <> x4
and
A7:
x3 <> x4
and
A8:
f . x1 = 1
and
A9:
for z being set st z in A & z <> x1 holds
f . z = 0
and
A10:
g . x2 = 1
and
A11:
for z being set st z in A & z <> x2 holds
g . z = 0
and
A12:
h . x3 = 1
and
A13:
for z being set st z in A & z <> x3 holds
h . z = 0
and
A14:
f1 . x4 = 1
and
A15:
for z being set st z in A & z <> x4 holds
f1 . z = 0
; for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
A16:
( f . x4 = 0 & g . x4 = 0 )
by A4, A6, A9, A11;
A17:
f1 . x3 = 0
by A7, A15;
A18:
f1 . x2 = 0
by A6, A15;
A19:
( f . x2 = 0 & h . x2 = 0 )
by A2, A5, A9, A13;
A20:
f1 . x1 = 0
by A4, A15;
A21:
( g . x1 = 0 & h . x1 = 0 )
by A2, A3, A11, A13;
A22:
h . x4 = 0
by A7, A13;
let h9 be Element of Funcs (A,REAL); ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take a = h9 . x1; ex b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take b = h9 . x2; ex c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take c = h9 . x3; ex d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
take d = h9 . x4; h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
A23:
( f . x3 = 0 & g . x3 = 0 )
by A3, A5, A9, A11;
now for x being Element of A holds h9 . x = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . xlet x be
Element of
A;
h9 . x = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . xA24:
(
x = x1 or
x = x2 or
x = x3 or
x = x4 )
by A1, ENUMSET1:def 2;
A25:
((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x2 =
(((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x2) + (((RealFuncExtMult A) . [d,f1]) . x2)
by FUNCSDOM:1
.=
((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x2) + (((RealFuncExtMult A) . [c,h]) . x2)) + (((RealFuncExtMult A) . [d,f1]) . x2)
by FUNCSDOM:1
.=
(((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2)) + (((RealFuncExtMult A) . [d,f1]) . x2)
by FUNCSDOM:1
.=
(((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2)) + (d * (f1 . x2))
by FUNCSDOM:4
.=
(((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (c * (h . x2))) + (d * (f1 . x2))
by FUNCSDOM:4
.=
(((((RealFuncExtMult A) . [a,f]) . x2) + (b * (g . x2))) + (c * (h . x2))) + (d * (f1 . x2))
by FUNCSDOM:4
.=
(((a * 0) + (b * 1)) + (c * 0)) + (d * 0)
by A10, A19, A18, FUNCSDOM:4
.=
h9 . x2
;
A26:
((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x4 =
(((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x4) + (((RealFuncExtMult A) . [d,f1]) . x4)
by FUNCSDOM:1
.=
((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x4) + (((RealFuncExtMult A) . [c,h]) . x4)) + (((RealFuncExtMult A) . [d,f1]) . x4)
by FUNCSDOM:1
.=
(((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (((RealFuncExtMult A) . [c,h]) . x4)) + (((RealFuncExtMult A) . [d,f1]) . x4)
by FUNCSDOM:1
.=
(((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (((RealFuncExtMult A) . [c,h]) . x4)) + (d * (f1 . x4))
by FUNCSDOM:4
.=
(((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (c * (h . x4))) + (d * (f1 . x4))
by FUNCSDOM:4
.=
(((((RealFuncExtMult A) . [a,f]) . x4) + (b * (g . x4))) + (c * (h . x4))) + (d * (f1 . x4))
by FUNCSDOM:4
.=
(((a * 0) + (b * 0)) + (c * 0)) + (d * 1)
by A14, A16, A22, FUNCSDOM:4
.=
h9 . x4
;
A27:
((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x3 =
(((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x3) + (((RealFuncExtMult A) . [d,f1]) . x3)
by FUNCSDOM:1
.=
((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x3) + (((RealFuncExtMult A) . [c,h]) . x3)) + (((RealFuncExtMult A) . [d,f1]) . x3)
by FUNCSDOM:1
.=
(((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3)) + (((RealFuncExtMult A) . [d,f1]) . x3)
by FUNCSDOM:1
.=
(((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3)) + (d * (f1 . x3))
by FUNCSDOM:4
.=
(((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (c * (h . x3))) + (d * (f1 . x3))
by FUNCSDOM:4
.=
(((((RealFuncExtMult A) . [a,f]) . x3) + (b * (g . x3))) + (c * (h . x3))) + (d * (f1 . x3))
by FUNCSDOM:4
.=
(((a * 0) + (b * 0)) + (c * 1)) + (d * 0)
by A12, A23, A17, FUNCSDOM:4
.=
h9 . x3
;
((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x1 =
(((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))) . x1) + (((RealFuncExtMult A) . [d,f1]) . x1)
by FUNCSDOM:1
.=
((((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))) . x1) + (((RealFuncExtMult A) . [c,h]) . x1)) + (((RealFuncExtMult A) . [d,f1]) . x1)
by FUNCSDOM:1
.=
(((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1)) + (((RealFuncExtMult A) . [d,f1]) . x1)
by FUNCSDOM:1
.=
(((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1)) + (d * (f1 . x1))
by FUNCSDOM:4
.=
(((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (c * (h . x1))) + (d * (f1 . x1))
by FUNCSDOM:4
.=
(((((RealFuncExtMult A) . [a,f]) . x1) + (b * (g . x1))) + (c * (h . x1))) + (d * (f1 . x1))
by FUNCSDOM:4
.=
(((a * 1) + (b * 0)) + (c * 0)) + (d * 0)
by A8, A21, A20, FUNCSDOM:4
.=
h9 . x1
;
hence
h9 . x = ((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))) . x
by A24, A25, A27, A26;
verum end;
hence
h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
by FUNCT_2:63; verum