reserve E, F, G,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 Th31:
  for E,F be RealNormSpace,
      Z be Subset of E,
      g be PartFunc of E,F,
      f be PartFunc of E,R_NormSpace_of_BoundedLinearOperators(E,F)
   st (g|Z) `| Z = f|Z
  holds
    for i be Nat holds diff(g,i+1,Z) = diff(f,i,Z)
proof
  let E,F be RealNormSpace,
      Z be Subset of E,
      g be PartFunc of E,F,
      f be PartFunc of E,R_NormSpace_of_BoundedLinearOperators(E,F);
  assume A1: (g|Z) `| Z = f|Z;

  defpred P[Nat] means
  diff(g,$1+1,Z) = diff(f,$1,Z);

  diff(g,0+1,Z)
   = (g|Z) `| Z by NDIFF_6:11
  .= diff(f,0,Z) by A1,NDIFF_6:11;
  then A2: P[0];

  A3: for i be Nat st P[i] holds P[i+1]
  proof
    let i be Nat;
    assume A4: P[i];
    thus diff(g,(i+1)+1,Z)
     = diff(g,i+1,Z) `| Z by NDIFF_6:13
    .= diff(f,i,Z) `| Z by A4,Th30
    .= diff(f,i+1,Z) by NDIFF_6:13;
  end;
  for i be Nat holds P[i] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
