
theorem Th3:
  for X being RealLinearSpace-Sequence,
      i be Element of dom X,
      x be Element of product X,
      r be Element of X.i,
      F be Function
  st F = reproj(i,x).r
  holds F.i = r
  proof
    let X be RealLinearSpace-Sequence,
    i be Element of dom X,
    x be Element of product X,
    r be Element of X.i,
    F be Function;
    assume
    A1: F = reproj(i,x).r;
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 Def20;
    thus F.i = r by A1,A2,A3,A4,Th1;
  end;
