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 Th10:
  for X,Y,Z be RealNormSpace,
          V be Subset of Y,
          g be PartFunc of Y,Z,
          I be LinearOperator of X,Y
   st I is one-to-one onto isometric
    & g is_differentiable_on V
  holds
      g`|V is_continuous_on V
    iff
      (g*I)`| (I"V) is_continuous_on (I"V)
  proof
    let X,Y,Z be RealNormSpace,
            V be Subset of Y,
            g be PartFunc of Y,Z,
            I be LinearOperator of X,Y;

    assume that
    A1: I is one-to-one onto isometric and
    A2: g is_differentiable_on V;

    consider J be LinearOperator of Y,X such that
    A3: J = I" & J is one-to-one onto isometric by A1,NDIFF_7:9;

    set f = g * I;
    set U = I"(V);
    A4: dom J = the carrier of Y by FUNCT_2:def 1;
    A5: f is_differentiable_on U by A1,A2,NDIFF_7:29;
    A6: dom(f`|U) = U by A5,NDIFF_1:def 9;
    A7: dom(g`|V) = V by A2,NDIFF_1:def 9;
    set F = f`|U;
    set G = g`|V;

    A8: G is_continuous_on V implies F is_continuous_on U
    proof
      assume
      A10: G is_continuous_on V;

      for x0 be Point of X, e be Real st x0 in U & 0 < e
      ex d be Real
      st 0 < d
       & for x1 be Point of X st x1 in U & ||. x1 - x0 .|| < d
         holds ||. F/.x1 - F/.x0 .|| < e
      proof
        let x0 be Point of X,
        e be Real;
        assume
        A11: x0 in U & 0 < e;
        consider y0 be Point of Y such that
        A12: x0 = J.y0 by A3,FUNCT_2:113;
        A13: I.x0 = y0 by A1,A3,A12,FUNCT_1:35; then
        A14: y0 in V by A11,FUNCT_2:38; then
        consider d be Real such that
        A15: 0 < d &
             for y1 be Point of Y st y1 in V & ||. y1 - y0 .|| < d
             holds ||. G/.y1 - G/.y0 .|| < e by A10,A11,NFCONT_1:19;
        take d;
        thus 0 < d by A15;

        thus for x1 be Point of X st x1 in U & ||. x1 - x0 .|| < d
             holds ||. F/.x1 - F/.x0 .|| < e
        proof
          let x1 be Point of X;
          assume
          A16: x1 in U & ||. x1 - x0 .|| < d;
          consider y1 be Point of Y such that
          A17: x1 = J.y1 by A3,FUNCT_2:113;
          A18: I.x1 = y1 by A1,A3,A17,FUNCT_1:35; then
          A19: y1 in V by A16,FUNCT_2:38;

          ||. y1 - y0 .|| = ||. x1 - x0 .|| by A1,A13,A18; then
          A20: ||. G/.y1 - G/.y0 .|| < e by A15,A16,A19;
          A21: G/.y1 = G.y1 by A7,A19,PARTFUN1:def 6
                    .= (F/.x1)*J by A1,A2,A3,A17,A19,Th9;
          A22: G/.y0 = G.y0 by A7,A14,PARTFUN1:def 6
                    .= (F/.x0)*J by A1,A2,A3,A12,A14,Th9;
          reconsider dF = F/.x1 - F/.x0
            as Lipschitzian LinearOperator of X,Z by LOPBAN_1:def 9;
          reconsider dG = G/.y1 - G/.y0
            as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;

          now
            let v be VECTOR of Y;
            A23: ((F/.x1 - F/.x0)*J).v
             = (F/.x1 - F/.x0).(J.v) by A4,FUNCT_1:13
            .= (F/.x1).(J.v) - (F/.x0).(J.v) by LOPBAN_1:40;
            A24: (G/.y1).v = (F/.x1).(J.v) by A4,A21,FUNCT_1:13;
            (G/.y0).v = (F/.x0).(J.v) by A4,A22,FUNCT_1:13;
            hence (dF*J).v = dG.v by A23,A24,LOPBAN_1:40;
          end; then
          dF*J = dG;
          hence ||. F/.x1 - F/.x0 .|| < e by A3,A20,NDIFF_7:22;
        end;
      end;
      hence thesis by A6,NFCONT_1:19;
    end;

    F is_continuous_on U implies G is_continuous_on V
    proof
      assume
      A25: F is_continuous_on U;

      for y0 be Point of Y,e be Real st y0 in V & 0 < e
      ex d be Real
      st 0 < d
       & for y1 be Point of Y st y1 in V & ||. y1 - y0 .|| < d
         holds ||. G/.y1 - G/.y0 .|| < e
      proof
        let y0 be Point of Y,
        e be Real;
        assume
        A26: y0 in V & 0<e;
        set x0 = J.y0;

        I.x0 = y0 by A1,A3,FUNCT_1:35; then
        x0 in U by A26,FUNCT_2:38; then
        consider d be Real such that
        A27: 0 < d
           & for x1 be Point of X st x1 in U & ||. x1 - x0 .|| < d
             holds ||. F/.x1 - F/.x0 .|| < e by A25,A26,NFCONT_1:19;

        take d;
        thus 0 < d by A27;
        let y1 be Point of Y;

        assume
        A28: y1 in V & ||. y1 - y0 .|| < d;

        set x1 = J.y1;
        I.x1 = y1 by A1,A3,FUNCT_1:35; then
        A29: x1 in U by A28,FUNCT_2:38;
        ||. x1 - x0 .|| = ||. y1 - y0 .|| by A3; then
        A30: ||. F/.x1 - F/.x0 .|| < e by A27,A28,A29;
        A31: G/.y1 = G.y1 by A7,A28,PARTFUN1:def 6
                  .= (F/.x1)*J by A1,A2,A3,A28,Th9;
        A32: G/.y0 = G.y0 by A7,A26,PARTFUN1:def 6
                  .= (F/.x0)*J by A1,A2,A3,A26,Th9;
        reconsider dF = F/.x1 - F/.x0
          as Lipschitzian LinearOperator of X,Z by LOPBAN_1:def 9;
        reconsider dG = G/.y1 - G/.y0
          as Lipschitzian LinearOperator of Y,Z by LOPBAN_1:def 9;

        now
          let v be VECTOR of Y;
          A33: ((F/.x1 - F/.x0)*J).v
           = (F/.x1 - F/.x0).(J.v) by A4,FUNCT_1:13
          .= (F/.x1).(J.v) - (F/.x0).(J.v) by LOPBAN_1:40;
          A34: (G/.y1).v = (F/.x1).(J.v) by A4,A31,FUNCT_1:13;
          (G/.y0).v = (F/.x0).(J.v) by A4,A32,FUNCT_1:13;
          hence (dF*J).v = dG.v by A33,A34,LOPBAN_1:40;
        end; then
        dF*J = dG;
        hence ||. G/.y1 - G/.y0 .|| < e by A3,A30,NDIFF_7:22;
      end;
      hence thesis by A7,NFCONT_1:19;
    end;
    hence thesis by A8;
  end;
