reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem REP2:
  for E,F be RealNormSpace,
    s1,s2 be Point of [:E,F:]
  st s1`1 = s2`1
  holds reproj2 s1 = reproj2 s2
  proof
    let E,F be RealNormSpace,
      s1,s2 be Point of [:E,F:];
    assume
    A1: s1`1 = s2`1;
    now
      let r be Element of F;
      thus (reproj2 s1) . r = [(s2 `1),r ] by A1,NDIFF_7:def 2
        .=(reproj2 s2) . r by NDIFF_7:def 2;
    end;
    hence thesis by FUNCT_2:63;
  end;
