 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,G be RealNormSpace,
      L be Point of R_NormSpace_of_BoundedLinearOperators([:E,F:],G)
  holds
    ex L1 be Point of R_NormSpace_of_BoundedLinearOperators(E,G),
       L2 be Point of R_NormSpace_of_BoundedLinearOperators(F,G)
    st ( for x be Point of E,
             y be Point of F
         holds L. [x,y] = L1.x + L2.y )
     & ( for x be Point of E holds L1.x = L. [x,0.F] )
     & ( for y be Point of F holds L2.y = L. [0.E,y] )
      & ||.L.|| <= ||.L1.|| + ||.L2.||
      & ||.L1.|| <= ||.L.|| & ||.L2.|| <= ||.L.||
  proof
    let E,F,G be RealNormSpace,
        LP be Point of R_NormSpace_of_BoundedLinearOperators([:E,F:],G);
    reconsider L = LP as Lipschitzian LinearOperator of [:E,F:],G
      by LOPBAN_1:def 9;
    consider L1 be Lipschitzian LinearOperator of E,G,
             L2 be Lipschitzian LinearOperator of F,G such that
    A1: ( for x be Point of E,
              y be Point of F
          holds L. [x,y] = L1.x + L2.y ) and
    A2: ( for x be Point of E holds L1.x = L/. [x,0.F] ) and
    A3: ( for y be Point of F holds L2.y = L/. [0.E,y] ) by Th2;
    reconsider LP1=L1 as Point of R_NormSpace_of_BoundedLinearOperators(E,G)
      by LOPBAN_1:def 9;
    reconsider LP2=L2 as Point of R_NormSpace_of_BoundedLinearOperators(F,G)
      by LOPBAN_1:def 9;
    take LP1,LP2;
    thus for x be Point of E, y be Point of F
         holds LP. [x,y] = LP1.x + LP2.y by A1;
    thus for x be Point of E holds LP1.x = LP. [x,0.F]
    proof
      let x be Point of E;
      thus LP1. x = L/.[x,0.F] by A2
                 .= LP. [x,0.F];
    end;
    thus for y be Point of F holds LP2.y = LP. [0.E,y]
    proof
      let y be Point of F;
      thus LP2. y = L/.[0.E,y] by A3
                 .= LP. [0.E,y];
    end;
    A7: ||.LP.||
      = upper_bound (PreNorms(modetrans(LP,[:E,F:],G))) by LOPBAN_1:def 13
     .= upper_bound PreNorms(L) by LOPBAN_1:29;
    for t be Real st t in PreNorms(L) holds t <= ||.LP1.|| + ||.LP2.||
    proof
      let t be Real;
      assume t in PreNorms(L); then
      consider w be Point of [:E,F:] such that
      A8: t = ||. L.w .|| & ||.w.|| <= 1;
      consider x be Point of E,y be Point of F such that
      A9: w =[x,y] by PRVECT_3:18;
      ||.x.|| <= ||.w.|| by A9,NDIFF_8:21; then
      A10: ||.x.|| <= 1 by A8,XXREAL_0:2;
      ||.y.|| <= ||.w.|| by A9,NDIFF_8:21; then
      A11: ||.y.|| <= 1 by A8,XXREAL_0:2;
      L. w = L1.x + L2.y by A1,A9; then
      A12: ||. L.w .|| <= ||.L1.x.|| + ||.L2.y.|| by NORMSP_1:def 1;
      A13: ||.L1.x.|| <= ||.LP1.|| * ||.x.|| by LOPBAN_1:32;
      ||.LP1.|| * ||.x.|| <= ||.LP1.|| * 1 by A10,XREAL_1:64; then
      A14: ||.L1.x.|| <= ||.LP1.|| by A13,XXREAL_0:2;
      A15: ||.L2.y.|| <= ||.LP2.|| * ||.y.|| by LOPBAN_1:32;
      ||.LP2.|| * ||.y.|| <= ||.LP2.|| * 1 by A11,XREAL_1:64; then
      ||.L2.y.|| <= ||.LP2.|| by A15,XXREAL_0:2; then
      ||.L1.x.|| + ||.L2.y.|| <= ||.LP1.|| + ||.LP2.|| by A14,XREAL_1:7;
      hence thesis by A8,A12,XXREAL_0:2;
    end;
    hence ||.LP.|| <= ||.LP1.|| + ||.LP2.|| by A7,SEQ_4:45;
    A17: ||.LP1.||
       = upper_bound (PreNorms(modetrans(LP1,E,G))) by LOPBAN_1:def 13
      .= upper_bound PreNorms(L1) by LOPBAN_1:29;
    for t be Real st t in PreNorms(L1) holds t <= ||.LP.||
    proof
      let t be Real;
      assume t in PreNorms(L1); then
      consider x be Point of E such that
      A18: t = ||. L1.x .|| & ||.x.|| <= 1;
      reconsider w = [x,0.F] as Point of [:E,F:];
      A19: ||.w.|| <= 1 by A18,NDIFF_8:2;
      A20: L. w = L1.x + L2. (0.F) by A1
      .= L1.x + 0.G by LOPBAN_7:3
      .= L1.x by RLVECT_1:def 4;
      A21: ||. L.w .|| <= ||.LP.|| * ||.w.|| by LOPBAN_1:32;
      ||.LP.|| * ||.w.|| <= ||.LP.|| * 1 by A19,XREAL_1:64;
      hence t <= ||.LP.|| by A18,A20,A21,XXREAL_0:2;
    end;
    hence ||.LP1.|| <= ||.LP.|| by A17,SEQ_4:45;
    A23: ||.LP2.||
       = upper_bound (PreNorms(modetrans(LP2,F,G))) by LOPBAN_1:def 13
      .= upper_bound PreNorms(L2) by LOPBAN_1:29;
    for t be Real st t in PreNorms(L2) holds t <= ||.LP.||
    proof
      let t be Real;
      assume t in PreNorms(L2); then
      consider x be Point of F such that
      A24: t = ||. L2.x .|| & ||.x.|| <= 1;
      reconsider w = [0.E,x] as Point of [:E,F:];
      A25: ||.x.|| = ||.w.|| by NDIFF_8:3;
      A26: L. w = L1.(0.E) + L2.x by A1
      .= 0.G + L2.x by LOPBAN_7:3
      .= L2.x by RLVECT_1:def 4;
      A27: ||. L.w .|| <= ||.LP.|| * ||.w.|| by LOPBAN_1:32;
      ||.LP.|| * ||.w.|| <= ||.LP.|| * 1 by A24,A25,XREAL_1:64;
      hence t <= ||.LP.|| by A24,A26,A27,XXREAL_0:2;
    end;
    hence ||.LP2.|| <= ||.LP.|| by A23,SEQ_4:45;
  end;
