
theorem Th18:
  for X be RealNormSpace,
      s be Point of product <*X*>,
      i be Element of dom <*X*>
  holds reproj(i,s) = IsoCPNrSP(X)
  proof
    let X be RealNormSpace,
        s be Point of product <*X*>,
        i be Element of dom <*X*>;

    A1: i = 1 by FINSEQ_1:90;
    A2: <*X*>.i = X by A1;

    for x be Element of X
    holds (reproj(i,s)).x = (IsoCPNrSP X).x
    proof
      let x be Element of X;

      s in the carrier of product <*X*>; then
      s in rng(IsoCPNrSP X) by FUNCT_2:def 3; then
      consider x0 be object such that
      A3: x0 in the carrier of X
        & s = (IsoCPNrSP X).x0 by FUNCT_2:11;

      reconsider x0 as Point of X by A3;
      A4: (IsoCPNrSP X).x0 = <*x0*> by Def2;
      dom s = Seg 1 by A3,A4,FINSEQ_1:38; then
      A5: i in dom s by A1;
      dom(s +* (i,x))
       = dom s by FUNCT_7:30
      .= Seg 1 by A3,A4,FINSEQ_1:38; then
      A6: len(s +* (i,x)) = 1 by FINSEQ_1:def 3;
      A7: (s +* (i,x)).1 = x by A1,A5,FUNCT_7:31;

      thus
      (reproj(i,s)).x
       = s +* (i,x) by A2,NDIFF_5:def 4
      .= <*x*> by A6,A7,FINSEQ_1:40
      .= (IsoCPNrSP X).x by Def2;
    end;
    hence reproj(i,s) = IsoCPNrSP(X) by A2;
  end;
