
theorem Th5:
  for X be RealBanachSpace, Y be RealNormSpace, T be Subset of
R_NormSpace_of_BoundedLinearOperators(X,Y) st for x be Point of X ex K be Real
 st 0 <= K & for f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
st f in T holds ||. f.x .|| <= K holds ex L be Real st 0 <= L & for f be
Point of R_NormSpace_of_BoundedLinearOperators(X,Y) st f in T holds ||.f.|| <=
  L
proof
  let X be RealBanachSpace, Y be RealNormSpace, T be Subset of
  R_NormSpace_of_BoundedLinearOperators(X,Y);
  assume
A1: for x be Point of X ex KTx be Real st 0 <= KTx & for f be
Point of R_NormSpace_of_BoundedLinearOperators(X,Y) st f in T holds ||. f.x .||
  <= KTx;
  per cases;
  suppose
A2: T <> {};
    deffunc S0(Point of X,Real) = Ball($1,$2);
    defpred P[Point of X,set] means $2={||. f.$1 .|| where f is Lipschitzian
    LinearOperator of X,Y :f in T};
A3: for x be Point of X ex y be Element of bool REAL st P[x,y]
    proof
      let x be Point of X;
      take y = {||. f.x .|| where f is Lipschitzian LinearOperator of X,Y :
      f in T};
      now
        let z be object;
        assume z in y;
        then
        ex f be Lipschitzian LinearOperator of X,Y st z=||. f.x .|| & f in T;
        hence z in REAL;
      end;
      hence thesis by TARSKI:def 3;
    end;
    ex Ta be Function of the carrier of X,bool REAL st for x be Element
 of X holds P[x,Ta.x] from FUNCT_2:sch 3(A3);
    then consider Ta be Function of X,bool REAL such that
A4: for x be Point of X holds Ta.x = {||. f.x .|| where f is Lipschitzian
    LinearOperator of X,Y :f in T};
    defpred P[Nat,set] means $2={x where x is Point of X: Ta.x is
    bounded_above & upper_bound(Ta.x) <= $1};
A5: for n be Element of NAT
        ex y be Element of bool the carrier of X st P [n,y]
    proof
      let n be Element of NAT;
      take y = {x where x is Point of X: Ta.x is bounded_above &
      upper_bound(Ta.x) <=
      n};
      now
        let z be object;
        assume z in y;
        then ex x be Point of X st z=x & Ta.x is bounded_above &
        upper_bound(Ta.x) <=
        n;
        hence z in the carrier of X;
      end;
      hence thesis by TARSKI:def 3;
    end;
    ex Xn be sequence of bool the carrier of X st for n be Element of
    NAT holds P[n,Xn.n] from FUNCT_2:sch 3(A5);
    then consider Xn be sequence of  bool the carrier of X such that
A6: for n be Element of NAT holds Xn.n = {x where x is Point of X: Ta
    .x is bounded_above & upper_bound(Ta.x) <= n};
    reconsider Xn as SetSequence of X;
A7: the carrier of X c= union rng Xn
    proof
      let x be object;
      assume x in the carrier of X;
      then reconsider x1=x as Point of X;
      consider KTx1 be Real such that
      0 <= KTx1 and
A8:   for f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y) st
      f in T holds ||. f.x1 .|| <= KTx1 by A1;
A9:   Ta.x1 = {||. f.x1 .|| where f is Lipschitzian LinearOperator of X,Y :f
      in T} by A4;
A10:  for p be Real st p in Ta.x1 holds p <= KTx1
      proof
        let p be Real;
        assume p in Ta.x1;
        then ex f be Lipschitzian LinearOperator of X,Y st p=||. f.x1 .|| &
        f in T by A9;
        hence thesis by A8;
      end;
   KTx1 is UpperBound of Ta.x1
      proof
        let p be ExtReal;
        assume p in Ta.x1;
        then ex f be Lipschitzian LinearOperator of X,Y st p=||. f.x1 .|| &
        f in T by A9;
        hence thesis by A8;
      end;
      then
A11:  Ta.x1 is bounded_above;
      consider n be Nat such that
A12:  KTx1 <n by SEQ_4:3;
A13:    n in NAT by ORDINAL1:def 12;
      consider f be object such that
A14:  f in T by A2,XBOOLE_0:def 1;
      reconsider f as Lipschitzian LinearOperator of X,Y by A14,LOPBAN_1:def 9;
      ||. f.x1 .|| in Ta.x by A9,A14;
      then upper_bound (Ta.x1) <= KTx1 by A10,SEQ_4:45;
      then upper_bound (Ta.x1) <= n by A12,XXREAL_0:2;
      then
      x1 in {z1 where z1 is Point of X : Ta.z1 is bounded_above &
      upper_bound (Ta.z1) <= n} by A11;
      then x1 in Xn.n by A6,A13;
      then x1 in Union Xn by PROB_1:12;
      hence thesis;
    end;
A15: for x be Point of X holds Ta.x is non empty
    proof
      let x be Point of X;
      consider f0 be object such that
A16:  f0 in T by A2,XBOOLE_0:def 1;
      reconsider f0 as Lipschitzian LinearOperator of X,Y
      by A16,LOPBAN_1:def 9;
      Ta.x = {||. f.x .|| where f is Lipschitzian LinearOperator of X,Y :f in
      T} by A4;
      then ||. f0.x .|| in Ta.x by A16;
      hence thesis;
    end;
A17: for n be Nat holds Xn.n is closed
    proof
      let n be Nat;
A18:  n in NAT by ORDINAL1:def 12;
      for s1 be sequence of X st rng s1 c= Xn.n & s1 is convergent holds
      lim s1 in Xn.n
      proof
        let s1 be sequence of X;
        assume that
A19:    rng s1 c= Xn.n and
A20:    s1 is convergent;
A21:    Ta.(lim s1) = {||. f.(lim s1) .|| where f is Lipschitzian
        LinearOperator of X,Y :f in T} by A4;
A22:    for y be Real st y in Ta.(lim s1) holds y <= n
        proof
          let y be Real;
          assume y in Ta.(lim s1);
          then consider f be Lipschitzian LinearOperator of X,Y such that
A23:      y=||. f.(lim s1) .|| and
A24:      f in T by A21;
A25:      f is_continuous_in lim s1 by Th4;
A26:      dom f=the carrier of X by FUNCT_2:def 1;
          then
A27:      rng s1 c= dom f by A19,XBOOLE_1:1;
          then
A28:      f/*s1 is convergent by A20,A25,NFCONT_1:def 5;
          for k be Nat holds ||. f/*s1 .||.k <= n
          proof
            let k be Nat;
A29:  k in NAT by ORDINAL1:def 12;
            ||. f/*s1 .||.k = ||. (f/*s1).k .|| by NORMSP_0:def 4;
            then
A30:        ||. f/*s1 .||.k =||. f/.(s1.k) .|| by A19,A26,FUNCT_2:109,A29
,XBOOLE_1:1;
            dom s1= NAT by FUNCT_2:def 1;
            then s1.k in rng s1 by FUNCT_1:3,A29;
            then s1.k in Xn.n by A19;
            then s1.k in {x where x is Point of X : Ta.x is bounded_above &
            upper_bound (Ta.x) <= n} by A6,A18;
            then consider x be Point of X such that
A31:        x=s1.k and
A32:        Ta.x is bounded_above and
A33:        upper_bound(Ta.x) <=n;
            Ta.x = {||. g.x .|| where g is Lipschitzian LinearOperator of X,Y
            :g in T} by A4;
            then ||. f.(s1.k) .|| in Ta.(s1.k) by A24,A31;
            then ||. f.(s1.k) .|| <= upper_bound (Ta.(s1.k))
            by A31,A32,SEQ_4:def 1;
            hence thesis by A30,A31,A33,XXREAL_0:2;
          end;
          then
A34:      for i being Nat st 0 <= i holds ||. f/*s1 .||.i <=n;
          f/.(lim s1) = lim (f/*s1) by A20,A25,A27,NFCONT_1:def 5;
          then lim (||. f/*s1 .||) = ||. f/.(lim s1) .|| by A28,LOPBAN_1:20;
          hence thesis by A23,A28,A34,NORMSP_1:23,RSSPACE2:5;
        end;
        then for y be ExtReal st y in Ta.(lim s1) holds y <= n;
        then
A35:    n is UpperBound of Ta.(lim s1) by XXREAL_2:def 1;
        Ta.(lim s1) is non empty by A15;
        then
A36:    upper_bound(Ta.(lim s1)) <= n by A22,SEQ_4:45;
A37:    Xn.n = {x where x is Point of X : Ta.x is bounded_above &
upper_bound (Ta
        .x) <= n} by A6,A18;
        Ta.(lim s1) is bounded_above by A35;
        hence thesis by A36,A37;
      end;
      hence thesis by NFCONT_1:def 3;
    end;
    consider f be object such that
A38: f in T by A2,XBOOLE_0:def 1;
    reconsider f as Lipschitzian LinearOperator of X,Y by A38,LOPBAN_1:def 9;
    union rng Xn is Subset of X by PROB_1:11;
    then union rng Xn = the carrier of X by A7,XBOOLE_0:def 10;
    then consider n0 be Nat, r be Real,
     x0 be Point of X such that
A39: 0 < r and
A40: S0(x0,r) c= Xn.n0 by A17,Th3;
A41:  n0 in NAT by ORDINAL1:def 12;
    ||.x0-x0.||=||.0.X.|| by RLVECT_1:5;
    then x0 in S0(x0,r) by A39;
    then x0 in Xn.n0 by A40;
    then x0 in {x1 where x1 is Point of X : Ta.x1 is bounded_above &
    upper_bound (Ta.
    x1) <= n0} by A6,A41;
    then consider xx1 be Point of X such that
A42: xx1=x0 and
A43: Ta.xx1 is bounded_above and
    upper_bound (Ta.xx1) <= n0;
    Ta.xx1 = {||. g.xx1 .|| where g is Lipschitzian LinearOperator of X,Y :g
    in T } by A4;
    then ||. f.x0 .|| in Ta.x0 by A42,A38;
    then ||. f.x0 .|| <= upper_bound(Ta.x0) by A42,A43,SEQ_4:def 1;
    then
A44: 0<= upper_bound (Ta.x0);
A45: for x be Point of X st x <> 0.X holds r/(2*||.x.||)*x+x0 in S0(x0,r)
    proof
      let x be Point of X;
      reconsider x1= (||.x.||")*x as Point of X;
A46:  ||.(r/2)*x1.|| = |.r/2.|*||.x1.|| by NORMSP_1:def 1;
      assume x <> 0.X;
      then
A47:  ||.x.|| <> 0 by NORMSP_0:def 5;
      ||. r/(2*||.x.||)*x+x0-x0.|| =||.r/(2*||.x.||)*x+(x0+-x0).|| by
RLVECT_1:def 3;
      then ||. r/(2*||.x.||)*x+x0-x0.|| =||.r/(2*||.x.||)*x+0.X.|| by
RLVECT_1:5;
      then ||. r/(2*||.x.||)*x+x0-x0.|| =||.r/(2*||.x.||)*x.|| by
RLVECT_1:def 4;
      then ||. r/(2*||.x.||)*x+x0-x0.|| =||.r/2/(||.x.||)*x.|| by XCMPLX_1:78;
      then
A48:  ||. r/(2*||.x.||)*x+x0-x0.|| =||.(r/2)*x1.|| by RLVECT_1:def 7;
A49:  |. ||.x.||".| =||.x.||" by ABSVALUE:def 1;
      ||.x1.|| = |. ||.x.||".|*||.x.|| by NORMSP_1:def 1;
      then ||.x1.|| = 1 by A47,A49,XCMPLX_0:def 7;
      then ||.(r/2)*x1.|| = r/2 by A39,A46,ABSVALUE:def 1;
      then
A50:  ||.x0-(r/(2*||.x.||)*x+x0).|| =r/2 by A48,NORMSP_1:7;
      r/2<r by A39,XREAL_1:216;
      hence thesis by A50;
    end;
    set M=upper_bound(Ta.x0);
    take KT=2*(M+n0)/r;
A51: for f be Lipschitzian LinearOperator of X,Y st f in T for x be Point of X
    st x in S0(x0,r) holds ||. f.x .|| <= n0
    proof
      let f be Lipschitzian LinearOperator of X,Y;
      assume
A52:  f in T;
A53:  n0 in NAT by ORDINAL1:def 12;
      let x be Point of X;
      assume x in S0(x0,r);
      then x in Xn.n0 by A40;
      then
      x in {x1 where x1 is Point of X : Ta.x1 is bounded_above &
      upper_bound (Ta.
      x1) <= n0} by A6,A53;
      then consider x1 be Point of X such that
A54:  x1=x and
A55:  Ta.x1 is bounded_above and
A56:  upper_bound(Ta.x1) <= n0;
      Ta.x1 = {||. g.x1 .|| where g is Lipschitzian LinearOperator of X,Y :g
      in T} by A4;
      then ||. f.x .|| in Ta.x by A52,A54;
      then ||. f.x .|| <= upper_bound (Ta.x) by A54,A55,SEQ_4:def 1;
      hence thesis by A54,A56,XXREAL_0:2;
    end;
A57: now
      let f be Lipschitzian LinearOperator of X,Y;
      assume
A58:  f in T;
A59:  for x be Point of X st x <> 0.X holds ||.f.x.|| <= KT*||.x.||
      proof
A60:  n0 in NAT by ORDINAL1:def 12;
        ||.x0-x0.||=||.0.X.|| by RLVECT_1:5;
        then x0 in S0(x0,r) by A39;
        then x0 in Xn.n0 by A40;
        then
A61:    x0 in {x1 where x1 is Point of X : Ta.x1 is bounded_above &
upper_bound(
        Ta.x1) <= n0} by A6,A60;
        set nr3=||.f.x0.||;
        let x be Point of X;
        set nrp1=r/(2*||.x.||);
        set nrp2=(2*||.x.||)/r;
        set nr1=||.f.(r/(2*||.x.||)*x)+f.x0.||;
        set nr2=||.f.(r/(2*||.x.||)*x).||;
        ||.-(f.x0).||=||.f.x0.|| by NORMSP_1:2;
        then
A62:    nr2-nr3<=||.f.(r/(2*||.x.||)*x)-(-f.x0).|| by NORMSP_1:8;
        assume
A63:    x <> 0.X;
        then
A64:    ||. f.(r/(2*||.x.||)*x+x0) .|| <= n0 by A51,A45,A58;
        consider x1 be Point of X such that
A65:    x1=x0 and
A66:    Ta.x1 is bounded_above and
        upper_bound (Ta.x1) <= n0 by A61;
        Ta.x1 = {||. g.x1 .|| where g is Lipschitzian LinearOperator of X,Y :
        g in T } by A4;
        then ||. f.x0 .|| in Ta.x0 by A58,A65;
        then ||. f.x0 .|| <= upper_bound (Ta.x0) by A65,A66,SEQ_4:def 1;
        then
A67:    nrp1*||.f.x.|| - M <= nrp1*||.f.x.|| - nr3 by XREAL_1:10;
        ||.x.|| <> 0 by A63,NORMSP_0:def 5;
        then
A68:    ||.x.|| >0;
        ||. f.(r/(2*||.x.||)*x).|| =||.(r/(2*||.x.||))*f.x.|| by LOPBAN_1:def 5
;
        then
||. f.(r/(2*||.x.||)*x).|| =|.r/(2*||.x.||).|*||.f.x.|| by NORMSP_1:def 1;
        then ||. f.(r/(2*||.x.||)*x).|| =(r/(2*||.x.||))*||.f.x.|| by A39,
ABSVALUE:def 1;
        then ||.f.(r/(2*||.x.||)*x)+f.x0.|| =||.f.(r/(2*||.x.||)*x+x0).|| & (
        r/(2*||.x.|| ))*||.f.x.||-nr3<=nr1 by A62,RLVECT_1:17,VECTSP_1:def 20;
        then (r/(2*||.x.||))*||.f.x.||-nr3<=n0 by A64,XXREAL_0:2;
        then nrp1*||.f.x.|| - M <= n0 by A67,XXREAL_0:2;
        then nrp1*||.f.x.|| + -M + M <= n0 + M by XREAL_1:6;
        then nrp2*(nrp1*||.f.x.||) <= nrp2*(n0+M) by A39,XREAL_1:64;
        then
A69:    nrp1*nrp2*||.f.x.|| <= nrp2*(n0+M);
        2*||.x.|| >0 by A68,XREAL_1:129;
        then 1*||.f.x.|| <= nrp2*(n0+M) by A39,A69,XCMPLX_1:112;
        hence thesis;
      end;
A70:  for x be Point of X holds ||.f.x.|| <= KT*||.x.||
      proof
        let x be Point of X;
        now
          assume
A71:      x = 0.X;
          then f.x = f.(0*0.X) by RLVECT_1:10;
          then f.x =0*f.(0.X) by LOPBAN_1:def 5;
          then
A72:      f.x =0.Y by RLVECT_1:10;
          ||.x.||= 0 by A71;
          hence thesis by A72;
        end;
        hence thesis by A59;
      end;
      thus for k be Real st k in {||.f.x1.|| where x1 is Point of X :
      ||.x1.|| <= 1 } holds k <= KT
      proof
        let k be Real;
        assume k in {||.f.x1.|| where x1 is Point of X : ||.x1.|| <= 1};
        then consider x be Point of X such that
A73:    k=||.f.x.|| & ||.x.|| <= 1;
        k <= KT*||.x.|| & KT*||.x.|| <=KT by A39,A44,A70,A73,XREAL_1:153;
        hence thesis by XXREAL_0:2;
      end;
    end;
    for f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y) st f in
    T holds ||.f.|| <= KT
    proof
      let f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
      reconsider f1=f as Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
      assume f in T;
      then
A74:  for k be Real st k in PreNorms(f1) holds k <= KT by A57;
      ||. f .|| = upper_bound PreNorms(f1) by LOPBAN_1:30;
      hence thesis by A74,SEQ_4:45;
    end;
    hence thesis by A39,A44;
  end;
  suppose
A75: T = {};
    take 0;
    thus thesis by A75;
  end;
end;
