
theorem Th22:
  for f, g being non-empty Function, i being object, A being set
  st A c= product f /\ product(f+*g) holds proj(f,i).:A = proj(f+*g,i).:A
proof
  let f, g be non-empty Function, i being object, A being set;
  assume A1: A c= product f /\ product(f+*g);
  for y being object holds y in proj(f,i).:A iff y in proj(f+*g,i).:A
  proof
    let y be object;
    hereby
      assume y in proj(f,i).:A;
      then consider x being object such that
      A2: x in dom proj(f,i) & x in A & y = proj(f,i).x by FUNCT_1:def 6;
      x in product(f+*g) by A1,A2,XBOOLE_0:def 4;
      then A4: x in dom proj(f+*g,i) by CARD_3:def 16;
      y = proj(f+*g,i).x by A2, A1, Th21;
      hence y in proj(f+*g,i).:A by A2, A4, FUNCT_1:def 6;
    end;
    assume y in proj(f+*g,i).:A;
    then consider x being object such that
    A5: x in dom proj(f+*g,i) & x in A & y = proj(f+*g,i).x by FUNCT_1:def 6;
    x in product f by A1,A5,XBOOLE_0:def 4;
    then A7: x in dom proj(f,i) by CARD_3:def 16;
    y = proj(f,i).x by A5, A1, Th21;
    hence y in proj(f,i).:A by A5, A7, FUNCT_1:def 6;
  end;
  hence thesis by TARSKI:2;
end;
