let x1, x2 be set ; for A being non empty set
for f, g being Element of Funcs (A,COMPLEX) st A = {x1,x2} & x1 <> x2 & ( for z being set st z in A holds
( ( z = x1 implies f . z = 1r ) & ( 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 = 1r ) ) ) holds
for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))
let A be non empty set ; for f, g being Element of Funcs (A,COMPLEX) st A = {x1,x2} & x1 <> x2 & ( for z being set st z in A holds
( ( z = x1 implies f . z = 1r ) & ( 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 = 1r ) ) ) holds
for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))
let f, g be Element of Funcs (A,COMPLEX); ( A = {x1,x2} & x1 <> x2 & ( for z being set st z in A holds
( ( z = x1 implies f . z = 1r ) & ( 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 = 1r ) ) ) implies for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g])) )
assume that
A1:
A = {x1,x2}
and
A2:
x1 <> x2
and
A3:
( ( for z being set st z in A holds
( ( z = x1 implies f . z = 1r ) & ( 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 = 1r ) ) ) )
; for h being Element of Funcs (A,COMPLEX) ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))
x2 in A
by A1, TARSKI:def 2;
then A4:
( f . x2 = 0c & g . x2 = 1r )
by A2, A3;
x1 in A
by A1, TARSKI:def 2;
then A5:
( f . x1 = 1r & g . x1 = 0c )
by A3;
let h be Element of Funcs (A,COMPLEX); ex a, b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))
reconsider x1 = x1, x2 = x2 as Element of A by A1, TARSKI:def 2;
take a = h . x1; ex b being Complex st h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))
take b = h . x2; h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))
now for x being Element of A holds h . x = ((ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))) . xlet x be
Element of
A;
h . x = ((ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))) . xA6:
(
x = x1 or
x = x2 )
by A1, TARSKI:def 2;
A7:
((ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))) . x2 =
(((ComplexFuncExtMult A) . [a,f]) . x2) + (((ComplexFuncExtMult A) . [b,g]) . x2)
by Th1
.=
(a * (f . x2)) + (((ComplexFuncExtMult A) . [b,g]) . x2)
by Th4
.=
0c + (b * 1r)
by A4, Th4
.=
h . x2
by COMPLEX1:def 4
;
((ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))) . x1 =
(((ComplexFuncExtMult A) . [a,f]) . x1) + (((ComplexFuncExtMult A) . [b,g]) . x1)
by Th1
.=
(a * (f . x1)) + (((ComplexFuncExtMult A) . [b,g]) . x1)
by Th4
.=
a + (b * 0c)
by A5, Th4, COMPLEX1:def 4
.=
h . x1
;
hence
h . x = ((ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))) . x
by A6, A7;
verum end;
hence
h = (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [b,g]))
by FUNCT_2:63; verum