
theorem Th1:
  for X be non empty non-empty FinSequence,
      x be Element of product X,
      i be Element of dom X,
      r be object st r in X.i holds
    (reproj (i,x).r ).i = r
  proof
    let X be non empty non-empty FinSequence,
        x be Element of product X,
        i be Element of dom X,
        r be object;
    assume r in X.i; then
    A1: reproj (i,x).r = x +* (i,r) by Def1;
    ex g being Function st x = g & dom g = dom X
     & for j being object st j in dom X holds
       g.j in X.j by CARD_3:def 5;
    hence (reproj (i,x).r).i = r by A1,FUNCT_7:31;
  end;
