let x, y, X be set ; :: thesis: ( x <> y implies ( Funcs X,{x,y}, bool X are_equipotent & card (Funcs X,{x,y}) = card (bool X) ) )
deffunc H1( Function) -> set = $1 " {x};
consider f being Function such that
A1: ( dom f = Funcs X,{x,y} & ( for g being Function st g in Funcs X,{x,y} holds
f . g = H1(g) ) ) from FUNCT_5:sch 1();
 assume A2: x <> y ; :: thesis: ( Funcs X,{x,y}, bool X are_equipotent & card (Funcs X,{x,y}) = card (bool X) )
thus Funcs X,{x,y}, bool X are_equipotent :: thesis: card (Funcs X,{x,y}) = card (bool X)
proof
deffunc H2( set ) -> set = x;
take f ; :: according to WELLORD2:def 4 :: thesis: ( f is one-to-one & proj1 f = Funcs X,{x,y} & proj2 f = bool X )
thus f is one-to-one :: thesis: ( proj1 f = Funcs X,{x,y} & proj2 f = bool X )
proof
let x1 be set ; :: according to FUNCT_1:def 8 :: thesis: for b1 being set holds
( not x1 in proj1 f or not b1 in proj1 f or not f . x1 = f . b1 or x1 = b1 )
let x2 be set ; :: thesis: ( not x1 in proj1 f or not x2 in proj1 f or not f . x1 = f . x2 or x1 = x2 )
assume that
A3: x1 in dom f and
A4: x2 in dom f and
A5: f . x1 = f . x2 ; :: thesis: x1 = x2
consider g2 being Function such that
A6: x2 = g2 and
A7: dom g2 = X and
A8: rng g2 c= {x,y} by A1, A4, FUNCT_2:def 2;
A9: f . x2 = g2 " {x} by A1, A4, A6;
consider g1 being Function such that
A10: x1 = g1 and
A11: dom g1 = X and
A12: rng g1 c= {x,y} by A1, A3, FUNCT_2:def 2;
A13: f . x1 = g1 " {x} by A1, A3, A10;
now
let z be set ; :: thesis: ( z in X implies g1 . z = g2 . z )
assume A14: z in X ; :: thesis: g1 . z = g2 . z
hence g1 . z = g2 . z by A15; :: thesis: verum
end;
hence x1 = x2 by A10, A11, A6, A7, FUNCT_1:9; :: thesis: verum
end;
thus dom f = Funcs X,{x,y} by A1; :: thesis: proj2 f = bool X
thus rng f c= bool X :: according to XBOOLE_0:def 10 :: thesis: bool X c= proj2 f
proof
let z be set ; :: according to TARSKI:def 3 :: thesis: ( not z in rng f or z in bool X )
assume z in rng f ; :: thesis: z in bool X
then consider t being set such that
A21: t in dom f and
A22: z = f . t by FUNCT_1:def 5;
consider g being Function such that
A23: t = g and
A24: dom g = X and
rng g c= {x,y} by A1, A21, FUNCT_2:def 2;
A25: g " {x} c= X by A24, RELAT_1:167;
z = g " {x} by A1, A21, A22, A23;
hence z in bool X by A25; :: thesis: verum
end;
deffunc H3( set ) -> set = y;
let z be set ; :: according to TARSKI:def 3 :: thesis: ( not z in bool X or z in proj2 f )
defpred S1[ set ] means $1 in z;
consider g being Function such that
A26: ( dom g = X & ( for t being set st t in X holds
( ( S1[t] implies g . t = H2(t) ) & ( not S1[t] implies g . t = H3(t) ) ) ) ) from PARTFUN1:sch 1();
 assume A27: z in bool X ; :: thesis: z in proj2 f
A28: g " {x} = z
proof
thus g " {x} c= z :: according to XBOOLE_0:def 10 :: thesis: z c= g " {x}
proof
let t be set ; :: according to TARSKI:def 3 :: thesis: ( not t in g " {x} or t in z )
assume A29: t in g " {x} ; :: thesis: t in z
then g . t in {x} by FUNCT_1:def 13;
then A30: g . t = x by TARSKI:def 1;
t in dom g by A29, FUNCT_1:def 13;
hence t in z by A2, A26, A30; :: thesis: verum
end;
let t be set ; :: according to TARSKI:def 3 :: thesis: ( not t in z or t in g " {x} )
assume A31: t in z ; :: thesis: t in g " {x}
then g . t = x by A27, A26;
then g . t in {x} by TARSKI:def 1;
hence t in g " {x} by A27, A26, A31, FUNCT_1:def 13; :: thesis: verum
end;
rng g c= {x,y}
proof
let t be set ; :: according to TARSKI:def 3 :: thesis: ( not t in rng g or t in {x,y} )
assume t in rng g ; :: thesis: t in {x,y}
then ex v being set st
( v in dom g & t = g . v ) by FUNCT_1:def 5;
then ( t = x or t = y ) by A26;
hence t in {x,y} by TARSKI:def 2; :: thesis: verum
end;
then A32: g in Funcs X,{x,y} by A26, FUNCT_2:def 2;
then f . g = g " {x} by A1;
hence z in proj2 f by A1, A32, A28, FUNCT_1:def 5; :: thesis: verum
end;
hence card (Funcs X,{x,y}) = card (bool X) by CARD_1:21; :: thesis: verum