reserve
  I for non empty set,
  F for associative Group-like multMagma-Family of I,
  i, j for Element of I;

theorem Th2:
  for x0 be set holds x0 in ProjSet(F,i) iff
  ex x be Function,g be Element of F.i st x = x0 & dom x = I & x.i = g
  & for j be Element of I st j <> i holds x.j = 1_F.j
  proof
    let x0 be set;
    defpred P[set] means
    ex g be Element of F.i st $1 = (1_(product F))+* (i,g);
    A1: now assume x0 in ProjSet(F,i); then
    consider g be Element of F.i such that
    A2: x0 = (1_(product F))+* (i,g) by Def1;
    reconsider x=x0 as Function by A2;
    take x,g;
    thus x=x0;
    thus dom x = I & x.i = g & for j be Element of I st j <> i holds
    x.j = 1_F.j by A2,Th1;
  end;
  now assume
    A3: ex x be Function,g be Element of F.i st x=x0 &
    dom x = I & x.i = g &
    for j be Element of I st j <> i holds x.j = 1_F.j;
    thus x0 in ProjSet(F,i)
    proof
      consider x be Function,g be Element of (F.i) such that
      A4: x=x0 & dom x = I & x.i = g &
      for j be Element of I st j <> i holds x.j = 1_F.j by A3;
      x = (1_(product F))+* (i,g) by Th1,A4;
      hence x0 in ProjSet(F,i) by Def1,A4;
    end;
  end;
  hence thesis by A1;
end;
