set C = [.0,1.];

defpred S_{1}[ Real, Real] means $1 > 0 ;

deffunc H_{1}( Element of [.0,1.], Element of [.0,1.]) -> Element of [.0,1.] = In (0,[.0,1.]);

deffunc H_{2}( Element of [.0,1.], Element of [.0,1.]) -> Element of [.0,1.] = In (1,[.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

( ( S_{1}[c,d] implies f . [c,d] = H_{1}(c,d) ) & ( not S_{1}[c,d] implies f . [c,d] = H_{2}(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

( ( S_{1}[c,d] implies f . [c,d] = H_{1}(c,d) ) & ( not S_{1}[c,d] implies f . [c,d] = H_{2}(c,d) ) )
;

take f ; :: thesis: for x, y being Element of [.0,1.] holds

( ( x = 0 implies f . (x,y) = 1 ) & ( x > 0 implies f . (x,y) = 0 ) )

A0: dom f = [:[.0,1.],[.0,1.]:] by FUNCT_2:def 1;

let a, b be Element of [.0,1.]; :: thesis: ( ( a = 0 implies f . (a,b) = 1 ) & ( a > 0 implies f . (a,b) = 0 ) )

AA: [a,b] in dom f by A0, ZFMISC_1:87;

( ( a > 0 implies f . (a,b) = 0 ) & ( a = 0 implies f . (a,b) = 1 ) )

defpred S

deffunc H

deffunc H

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

( ( S

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

( ( S

take f ; :: thesis: for x, y being Element of [.0,1.] holds

( ( x = 0 implies f . (x,y) = 1 ) & ( x > 0 implies f . (x,y) = 0 ) )

A0: dom f = [:[.0,1.],[.0,1.]:] by FUNCT_2:def 1;

let a, b be Element of [.0,1.]; :: thesis: ( ( a = 0 implies f . (a,b) = 1 ) & ( a > 0 implies f . (a,b) = 0 ) )

AA: [a,b] in dom f by A0, ZFMISC_1:87;

( ( a > 0 implies f . (a,b) = 0 ) & ( a = 0 implies f . (a,b) = 1 ) )

proof

hence
( ( a = 0 implies f . (a,b) = 1 ) & ( a > 0 implies f . (a,b) = 0 ) )
; :: thesis: verum
thus
( a > 0 implies f . (a,b) = 0 )
:: thesis: ( a = 0 implies f . (a,b) = 1 )

then f . [a,b] = H_{2}(a,b)
by A1, AA

.= 1 by XXREAL_1:1, SUBSET_1:def 8 ;

hence f . (a,b) = 1 ; :: thesis: verum

end;proof

assume
a = 0
; :: thesis: f . (a,b) = 1
assume
a > 0
; :: thesis: f . (a,b) = 0

then f . [a,b] = H_{1}(a,b)
by A1, AA

.= 0 by SUBSET_1:def 8, XXREAL_1:1 ;

hence f . (a,b) = 0 ; :: thesis: verum

end;then f . [a,b] = H

.= 0 by SUBSET_1:def 8, XXREAL_1:1 ;

hence f . (a,b) = 0 ; :: thesis: verum

then f . [a,b] = H

.= 1 by XXREAL_1:1, SUBSET_1:def 8 ;

hence f . (a,b) = 1 ; :: thesis: verum