set C = [.0,1.];
defpred S₁[ Real, Real] means $1 + $2 > 1;
deffunc H₁( Element of [.0,1.], Element of [.0,1.]) -> Element of [.0,1.] = In ((min ($1,$2)),[.0,1.]);
deffunc H₂( Element of [.0,1.], Element of [.0,1.]) -> Element of [.0,1.] = In (0,[.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 ; :: thesis: for a, b being Element of [.0,1.] holds
( ( a + b > 1 implies f . (a,b) = min (a,b) ) & ( a + b <= 1 implies f . (a,b) = 0 ) )
A0: dom f = [:[.0,1.],[.0,1.]:] by FUNCT_2:def 1;
let a, b be Element of [.0,1.]; :: thesis: ( ( a + b > 1 implies f . (a,b) = min (a,b) ) & ( a + b <= 1 implies f . (a,b) = 0 ) )
cc: ( min (a,b) = a or min (a,b) = b ) by XXREAL_0:15;
AA: [a,b] in dom f by A0, ZFMISC_1:87;
( ( a + b > 1 implies f . (a,b) = min (a,b) ) & ( a + b <= 1 implies f . (a,b) = 0 ) )

proof

thus ( a + b > 1 implies f . (a,b) = min (a,b) ) :: thesis: ( a + b <= 1 implies f . (a,b) = 0 )

proof

assume a + b > 1 ; :: thesis: f . (a,b) = min (a,b)
then f . [a,b] = H₁(a,b) by A1, AA
.= min (a,b) by SUBSET_1:def 8, cc ;
hence f . (a,b) = min (a,b) ; :: thesis: verum

end;

assume a + b <= 1 ; :: thesis: f . (a,b) = 0
then f . [a,b] = H₂(a,b) by A1, AA
.= 0 by XXREAL_1:1, SUBSET_1:def 8 ;
hence f . (a,b) = 0 ; :: thesis: verum

end;

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