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 Th6:
  for f be Lipschitzian BilinearOperator of E,F,G
  holds
    ex K be Real st 0 <= K
      &
    for z be Point of [:E,F:]
    holds
      ||. partdiff`1(f,z) .|| <= K * ||.z.||
    & ||. partdiff`2(f,z) .|| <= K * ||.z.||
  proof
    let f be Lipschitzian BilinearOperator of E,F,G;
    consider K be Real such that
    A1: 0 <= K
          &
        for z be Point of [:E,F:]
        holds
        ( for x be Point of E
          holds ||. (f * (reproj1 z)).x .|| <= K * ||.z`2.|| * ||.x.|| )
            &
        ( for y be Point of F
          holds ||. (f * (reproj2 z)).y .|| <= K * ||.z`1.|| * ||.y.|| )
          by Th3;

    take K;
    thus 0 <= K by A1;

    let z be Point of [:E,F:];
    reconsider L1 = f * (reproj1 z) as Lipschitzian LinearOperator of E,G
      by Th2;
    reconsider L2 = f * (reproj2 z) as Lipschitzian LinearOperator of F,G
      by Th2;
    A2: partdiff`1(f,z) = L1 by Th4;
    A3: partdiff`2(f,z) = L2 by Th4;
    A4: z = [z`1, z`2] by Th5;
    A5: ( for s be Real st s in PreNorms L2 holds s <= K * ||.z.|| )
        implies upper_bound PreNorms L2 <= K * ||.z.|| by SEQ_4:45;

    A6:
    now
      let t be VECTOR of F such that
      A7: ||.t.|| <= 1;
      A8: ||. L2.t .|| <= K * ||.z`1.|| * ||.t.|| by A1;
      0 <= ||.z `1.|| by NORMSP_1:4;
      then 0 <= K * ||.z `1.|| by A1,XREAL_1:127;
      then K * ||.z`1.|| * ||.t.|| <= K * ||.z`1.|| * 1 by A7,XREAL_1:64;
      then
      A9: ||. L2.t .|| <= K * ||.z`1.|| by A8,XXREAL_0:2;
      K * ||.z`1.|| <= K * ||.z.|| by A1,A4,LOPBAN_7:15,XREAL_1:64;
      hence ||. L2.t .|| <= K * ||.z.|| by A9,XXREAL_0:2;
    end;

    A10:
    now
      let r be Real;
      assume r in PreNorms L2;
      then ex t be VECTOR of F st r= ||.L2.t.|| & ||.t.|| <= 1;
      hence r <= K * ||.z.|| by A6;
    end;

    A11: ( for s be Real st s in PreNorms L1 holds s <= K * ||.z.|| )
        implies upper_bound PreNorms L1 <= K * ||.z.|| by SEQ_4:45;

    A12:
    now
      let t be VECTOR of E such that
      A13: ||.t.|| <= 1;
      A14: ||. L1.t .|| <= K * ||.z`2.|| * ||.t.|| by A1;
      0 <= ||.z `2.|| by NORMSP_1:4;
      then 0 <= K * ||.z `2.|| by A1,XREAL_1:127;
      then K * ||.z`2.|| * ||.t.|| <= K * ||.z`2.|| * 1 by A13,XREAL_1:64;
      then
      A15: ||. L1.t .|| <= K * ||.z`2.|| by A14,XXREAL_0:2;
      K * ||.z`2.|| <= K * ||.z.|| by A1,A4,LOPBAN_7:15,XREAL_1:64;
      hence ||. L1.t .|| <= K * ||.z.|| by A15,XXREAL_0:2;
    end;

    now
      let r be Real;
      assume r in PreNorms L1;
      then ex t be VECTOR of E st r=||.L1.t.|| & ||.t.|| <= 1;
      hence r <= K*||.z.|| by A12;
    end;
    hence thesis by A2,A3,A5,A10,A11,LOPBAN_1:30;
  end;
