
theorem Th9:
  for f being Function for i,A being set st [:A,{i}:] c= dom f
  holds pi((curry f).:A, i) = f.:[:A,{i}:]
proof
  let f be Function, i, A be set such that
A1: [:A,{i}:] c= dom f;
A2: i in {i} by TARSKI:def 1;
  thus pi((curry f).:A, i) c= f.:[:A,{i}:]
  proof
    let x be object;
    assume x in pi((curry f).:A, i);
    then consider g being Function such that
A3: g in (curry f).:A and
A4: x = g.i by CARD_3:def 6;
    consider a being object such that
    a in dom curry f and
A5: a in A and
A6: g = (curry f).a by A3,FUNCT_1:def 6;
A7: [a,i] in [:A, {i}:] by A2,A5,ZFMISC_1:def 2;
    then f.(a,i) in f.:[:A, {i}:] by A1,FUNCT_1:def 6;
    hence thesis by A1,A4,A6,A7,FUNCT_5:20;
  end;
  let x be object;
  assume x in f.:[:A,{i}:];
  then consider y being object such that
A8: y in dom f and
A9: y in [:A, {i}:] and
A10: x = f.y by FUNCT_1:def 6;
  consider y1,y2 being object such that
A11: y1 in A and
A12: y2 in {i} and
A13: y = [y1,y2] by A9,ZFMISC_1:def 2;
  reconsider g = (curry f).y1 as Function;
  y1 in dom curry f by A8,A13,FUNCT_5:19;
  then
A14: g in (curry f).:A by A11,FUNCT_1:def 6;
A15: y2 = i by A12,TARSKI:def 1;
  x = f.(y1,y2) by A10,A13;
  then x = g.i by A15,A8,A13,FUNCT_5:20;
  hence thesis by A14,CARD_3:def 6;
end;
