set C = [.0,1.];
defpred S1[ Real, Real] means $1 <= $2;
deffunc H1( Element of [.0,1.], Element of [.0,1.]) -> Element of [.0,1.] = In (1,[.0,1.]);
deffunc H2( Element of [.0,1.], Element of [.0,1.]) -> Element of [.0,1.] = In ($2,[.0,1.]);
ex f being Function of [:[.0,1.],[.0,1.]:],[.0,1.] st
for c, d being Element of [.0,1.] st [c,d] in dom f holds
( ( S1[c,d] implies f . [c,d] = H1(c,d) ) & ( not S1[c,d] implies f . [c,d] = H2(c,d) ) )
from SCHEME1:sch 21();
then consider f being Function of [:[.0,1.],[.0,1.]:],[.0,1.] such that
A1:
for c, d being Element of [.0,1.] st [c,d] in dom f holds
( ( S1[c,d] implies f . [c,d] = H1(c,d) ) & ( not S1[c,d] implies f . [c,d] = H2(c,d) ) )
;
take
f
; for x, y being Element of [.0,1.] holds
( ( x <= y implies f . (x,y) = 1 ) & ( x > y implies f . (x,y) = y ) )
A0:
dom f = [:[.0,1.],[.0,1.]:]
by FUNCT_2:def 1;
let a, b be Element of [.0,1.]; ( ( a <= b implies f . (a,b) = 1 ) & ( a > b implies f . (a,b) = b ) )
AA:
[a,b] in dom f
by A0, ZFMISC_1:87;
( ( a <= b implies f . (a,b) = 1 ) & ( a > b implies f . (a,b) = b ) )
proof
thus
(
a <= b implies
f . (
a,
b)
= 1 )
( a > b implies f . (a,b) = b )
assume
a > b
;
f . (a,b) = b
then f . [a,b] =
H2(
a,
b)
by A1, AA
.=
b
by SUBSET_1:def 8
;
hence
f . (
a,
b)
= b
;
verum
end;
hence
( ( a <= b implies f . (a,b) = 1 ) & ( a > b implies f . (a,b) = b ) )
; verum