reserve X,Y,Z,X1,X2,Y1,Y2 for set, x,y,z,t,x1,x2 for object,
  f,g,h,f1,f2,g1,g2 for Function;
reserve C,D,E for non empty set;
reserve c for Element of C,
  d for Element of D;
reserve f for Function of [:C,D:],E;

theorem Th61:
  curry' f is Function of D,Funcs(C,E)
proof
A1: dom f = [:C,D:] by FUNCT_2:def 1;
A2: rng curry' f c= Funcs(C,E)
  proof
A3: rng curry' f c= Funcs(C,rng f) by A1,Th28;
    let x be object;
    assume x in rng curry' f;
    then consider g being Function such that
A4: x = g and
A5: dom g = C and
A6: rng g c= rng f by A3,FUNCT_2:def 2;
    rng g c= E by A6,XBOOLE_1:1;
    then g is Function of C,E by A5,FUNCT_2:def 1,RELSET_1:4;
    hence thesis by A4,FUNCT_2:8;
  end;
  dom curry' f = D by A1,Th17;
  hence thesis by A2,FUNCT_2:def 1,RELSET_1:4;
end;
