set C = [.0,1.];
defpred S₁[ Real, Real] means ( $1 = $2 & $2 = 0 );
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 to_power $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
( ( 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 x, y being Element of [.0,1.] holds
( ( x = y & y = 0 implies f . (x,y) = 1 ) & ( ( x > 0 or y > 0 ) implies f . (x,y) = y to_power x ) )
A0: dom f = [:[.0,1.],[.0,1.]:] by FUNCT_2:def 1;
let a, b be Element of [.0,1.]; :: thesis: ( ( a = b & b = 0 implies f . (a,b) = 1 ) & ( ( a > 0 or b > 0 ) implies f . (a,b) = b to_power a ) )
AA: [a,b] in dom f by A0, ZFMISC_1:87;
( ( a = b & b = 0 implies f . (a,b) = 1 ) & ( ( a > 0 or b > 0 ) implies f . (a,b) = b to_power a ) )

proof

thus ( a = b & b = 0 implies f . (a,b) = 1 ) :: thesis: ( ( a > 0 or b > 0 ) implies f . (a,b) = b to_power a )

proof

assume ( a = b & b = 0 ) ; :: thesis: f . (a,b) = 1
then f . [a,b] = H₁(a,b) by A1, AA
.= 1 by SUBSET_1:def 8, XXREAL_1:1 ;
hence f . (a,b) = 1 ; :: thesis: verum

end;

assume X0: ( a > 0 or b > 0 ) ; :: thesis: f . (a,b) = b to_power a
then X1: b to_power a in [.0,1.] by PowerIn01;
not S₁[a,b] by X0;
then f . [a,b] = H₂(a,b) by A1, AA
.= b to_power a by SUBSET_1:def 8, X1 ;
hence f . (a,b) = b to_power a ; :: thesis: verum

end;

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