
theorem Th4:
  for X being RealNormSpace-Sequence,
    i,j be Element of dom X,
      x be Element of product X,
      r be Element of X.i,
    F,s be Function
  st F = reproj (i,x).r & x = s & i <> j
  holds F.j = s.j
  proof
    let X be RealNormSpace-Sequence,
      i,j be Element of dom X,
        x be Element of product X,
        r be Element of X.i,
      F,s be Function;
    assume
    A1: F = reproj(i,x).r & x = s & i <> j;
    A2: dom carr X = dom X by LemmaX;
    A3: (carr X).i = the carrier of (X.i) by PRVECT_1:def 11;
    consider x0 be Element of product carr X such that
    A4: x0 = x & reproj (i,x) = reproj(i,x0) by Def2;
    thus F.j = s.j by A1,A2,A3,A4,Th2;
  end;
