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 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 h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (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 ; :: thesis: 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 h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (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; :: thesis: ( 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 h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (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} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 )
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 ) )
and
A5:
( f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds
f1 . z = 0 ) )
; :: thesis: for h' being Element of Funcs A,REAL ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
A6:
( f . x1 = 1 & f . x2 = 0 & f . x3 = 0 & f . x4 = 0 & g . x1 = 0 & g . x2 = 1 & g . x3 = 0 & g . x4 = 0 & h . x1 = 0 & h . x2 = 0 & h . x3 = 1 & h . x4 = 0 & f1 . x1 = 0 & f1 . x2 = 0 & f1 . x3 = 0 & f1 . x4 = 1 )
by A1, A2, A3, A4, A5;
let h' be Element of Funcs A,REAL ; :: thesis: ex a, b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
take a = h' . x1; :: thesis: ex b, c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
take b = h' . x2; :: thesis: ex c, d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
take c = h' . x3; :: thesis: ex d being Real st h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
take d = h' . x4; :: thesis: h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
now let x be
Element of
A;
:: thesis: h' . x = ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])) . xA7:
(
x = x1 or
x = x2 or
x = x3 or
x = x4 )
by A1, ENUMSET1:def 2;
A8:
((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:10
.=
((((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) . x1) + (((RealFuncExtMult A) . [c,h]) . x1)) + (((RealFuncExtMult A) . [d,f1]) . x1)
by FUNCSDOM:10
.=
(((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1)) + (((RealFuncExtMult A) . [d,f1]) . x1)
by FUNCSDOM:10
.=
(((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (((RealFuncExtMult A) . [c,h]) . x1)) + (d * (f1 . x1))
by FUNCSDOM:15
.=
(((((RealFuncExtMult A) . [a,f]) . x1) + (((RealFuncExtMult A) . [b,g]) . x1)) + (c * (h . x1))) + (d * (f1 . x1))
by FUNCSDOM:15
.=
(((((RealFuncExtMult A) . [a,f]) . x1) + (b * (g . x1))) + (c * (h . x1))) + (d * (f1 . x1))
by FUNCSDOM:15
.=
(((a * 1) + (b * 0 )) + (c * 0 )) + (d * 0 )
by A6, FUNCSDOM:15
.=
h' . x1
;
A9:
((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:10
.=
((((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) . x2) + (((RealFuncExtMult A) . [c,h]) . x2)) + (((RealFuncExtMult A) . [d,f1]) . x2)
by FUNCSDOM:10
.=
(((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2)) + (((RealFuncExtMult A) . [d,f1]) . x2)
by FUNCSDOM:10
.=
(((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (((RealFuncExtMult A) . [c,h]) . x2)) + (d * (f1 . x2))
by FUNCSDOM:15
.=
(((((RealFuncExtMult A) . [a,f]) . x2) + (((RealFuncExtMult A) . [b,g]) . x2)) + (c * (h . x2))) + (d * (f1 . x2))
by FUNCSDOM:15
.=
(((((RealFuncExtMult A) . [a,f]) . x2) + (b * (g . x2))) + (c * (h . x2))) + (d * (f1 . x2))
by FUNCSDOM:15
.=
(((a * 0 ) + (b * 1)) + (c * 0 )) + (d * 0 )
by A6, FUNCSDOM:15
.=
h' . x2
;
A10:
((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:10
.=
((((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) . x3) + (((RealFuncExtMult A) . [c,h]) . x3)) + (((RealFuncExtMult A) . [d,f1]) . x3)
by FUNCSDOM:10
.=
(((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3)) + (((RealFuncExtMult A) . [d,f1]) . x3)
by FUNCSDOM:10
.=
(((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (((RealFuncExtMult A) . [c,h]) . x3)) + (d * (f1 . x3))
by FUNCSDOM:15
.=
(((((RealFuncExtMult A) . [a,f]) . x3) + (((RealFuncExtMult A) . [b,g]) . x3)) + (c * (h . x3))) + (d * (f1 . x3))
by FUNCSDOM:15
.=
(((((RealFuncExtMult A) . [a,f]) . x3) + (b * (g . x3))) + (c * (h . x3))) + (d * (f1 . x3))
by FUNCSDOM:15
.=
(((a * 0 ) + (b * 0 )) + (c * 1)) + (d * 0 )
by A6, FUNCSDOM:15
.=
h' . x3
;
((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:10
.=
((((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])) . x4) + (((RealFuncExtMult A) . [c,h]) . x4)) + (((RealFuncExtMult A) . [d,f1]) . x4)
by FUNCSDOM:10
.=
(((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (((RealFuncExtMult A) . [c,h]) . x4)) + (((RealFuncExtMult A) . [d,f1]) . x4)
by FUNCSDOM:10
.=
(((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (((RealFuncExtMult A) . [c,h]) . x4)) + (d * (f1 . x4))
by FUNCSDOM:15
.=
(((((RealFuncExtMult A) . [a,f]) . x4) + (((RealFuncExtMult A) . [b,g]) . x4)) + (c * (h . x4))) + (d * (f1 . x4))
by FUNCSDOM:15
.=
(((((RealFuncExtMult A) . [a,f]) . x4) + (b * (g . x4))) + (c * (h . x4))) + (d * (f1 . x4))
by FUNCSDOM:15
.=
(((a * 0 ) + (b * 0 )) + (c * 0 )) + (d * 1)
by A6, FUNCSDOM:15
.=
h' . x4
;
hence
h' . 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 A7, A8, A9, A10;
:: thesis: verum end;
hence
h' = (RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncAdd A) . ((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g])),((RealFuncExtMult A) . [c,h])),((RealFuncExtMult A) . [d,f1])
by FUNCT_2:113; :: thesis: verum