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
  for E,F be RealNormSpace,
    s1,s2 be Point of [:E,F:]
  st s1`2 = s2`2
  holds reproj1 s1 = reproj1 s2
  proof
    let E,F be RealNormSpace,
      s1,s2 be Point of [:E,F:];
    assume
    A1: s1`2 = s2`2;
    now
      let r be Element of E;
      thus (reproj1 s1) . r = [r,(s2 `2) ] by A1,NDIFF_7:def 1
        .= (reproj1 s2) . r by NDIFF_7:def 1;
    end;
    hence thesis by FUNCT_2:63;
  end;
