
theorem Th4X:
  for X being RealNormSpace-Sequence,
      i be Element of dom X,
      x be Element of product X,
      s be Function
  st x = s
  holds reproj(i,x).(s.i)= x
  proof
    let X be RealNormSpace-Sequence,
        i be Element of dom X,
        x be Element of product X,
        s be Function;
    assume
    A1: x = s;
    A2: dom carr X = dom X by LemmaX;
    consider x0 be Element of product carr X such that
    A4: x0 = x & reproj(i,x) = reproj(i,x0) by Def2;
    thus reproj (i,x).(s.i) = x by A1,A2,A4,Th2X;
  end;
