reserve X, Y for RealNormSpace;

theorem Th15:
  for X be RealBanachSpace, Y be RealNormSpace, T be Lipschitzian
LinearOperator of X,Y, r be Real, BX1 be Subset of LinearTopSpaceNorm X,
TBX1,BYr be Subset of LinearTopSpaceNorm Y st r>0 & BYr=Ball(0.Y,r ) & TBX1=T.:
  Ball(0.X,1) & BYr c= Cl (TBX1) holds BYr c= TBX1
proof
  let X be RealBanachSpace, Y be RealNormSpace,
  T be Lipschitzian LinearOperator of
  X,Y, r be Real, BX1 be Subset of LinearTopSpaceNorm X, TBX1,BYr be
  Subset of LinearTopSpaceNorm Y;
  assume that
A1: r>0 and
A2: BYr=Ball(0.Y,r ) and
A3: TBX1=T.: Ball(0.X,1) and
A4: BYr c= Cl (TBX1);
A5: for x be Point of X,y be Point of Y, TB1, BYsr be Subset of
  LinearTopSpaceNorm Y, s be Real st s >0 & TB1=T.: Ball(x,s) & y=T.x &
  BYsr =Ball(y,s*r) holds BYsr c= Cl (TB1)
  proof
    reconsider TB01 = T.: Ball(0.X,1) as Subset of Y;
    let x be Point of X, y be Point of Y, TB1, BYsr be Subset of
    LinearTopSpaceNorm Y, s be Real;
    assume that
A6: s >0 and
A7: TB1=T.: Ball(x,s) and
A8: y=T.x and
A9: BYsr =Ball(y,s*r);
    reconsider s1 = s as non zero Real by A6;
    reconsider y1=y as Point of LinearTopSpaceNorm Y by NORMSP_2:def 4;
A10: Ball(y,s*r) =y + Ball(0.Y,s*r) by Th2;
    reconsider TB0xs = T.: Ball(x,s) as Subset of Y;
    reconsider TB0s = T.: Ball(0.X,s) as Subset of Y;
    Ball(x,s) = x+Ball(0.X,s) by Th2;
    then
A11: y+TB0s = TB0xs by A8,Th6;
    s1*BYr c= s1*Cl (TBX1) by A4,CONVEX1:39;
    then s1*BYr c= Cl (s1*TBX1) by RLTOPSP1:52;
    then y1+s1*BYr c= y1+Cl (s1*TBX1) by RLTOPSP1:8;
    then
A12: y1+s1*BYr c= Cl (y1+s1*TBX1) by RLTOPSP1:38;
    Ball(0.Y,s*r)=s1*Ball(0.Y,r) by A6,Th3;
    then Ball(0.Y,s*r) = s1*BYr by A2,Th9;
    then
A13: y1+s1*BYr=BYsr by A9,A10,Th8;
A14: s1*Ball(0.X,1)= Ball(0.X,s1*1) by A6,Th3;
    s1*TB01 =s1* TBX1 by A3,Th9;
    hence thesis by A7,A12,A13,A11,A14,Th5,Th8;
  end;
A15: for s0 be Real st s0 >0 holds Ball(0.Y,r ) c= T.: Ball(0.X,1+s0)
  proof
    let s0 be Real;
    assume
A16: s0 >0;
    now
      let z be object;
      assume
A17:  z in Ball(0.Y,r);
      then reconsider y =z as Point of Y;
      consider s1 be Real such that
A18:  0 < s1 and
A19:  s1 < s0 by A16,XREAL_1:5;
      set a=s1/(1+s1);
      set e= a GeoSeq;
A20:  a<1 by A18,XREAL_1:29,191;
      then
A21:  |.a.| < 1 by A18,ABSVALUE:def 1;
      then
A22:  e is summable by SERIES_1:24;
      defpred P[Nat,Point of X,Point of X, Point of X] means $3 in
Ball($2,e.$1) & ||.T.$3-y.|| < e.($1+1)*r implies $4 in Ball($3,e.($1+1)) & ||.
      T.$4-y.|| < e.($1+2)*r;
      reconsider B0 =Ball(y,e.1*r) as Subset of LinearTopSpaceNorm Y by
NORMSP_2:def 4;
A23:  0<a by A18,XREAL_1:139;
      then e.1=a|^1 & 0<a|^1 by PREPOWER:def 1;
      then 0< e.1*r by A1,XREAL_1:129;
      then ||.y-y.|| < e.1*r by NORMSP_1:6;
      then B0 is open & y in B0 by NORMSP_2:23;
      then Ball(y,e.1*r) meets TBX1 by A2,A4,A17,PRE_TOPC:def 7;
      then consider s be object such that
A24:  s in Ball(y,e.1*r) and
A25:  s in T.: Ball(0.X,1) by A3,XBOOLE_0:3;
      consider xn1 be object such that
A26:  xn1 in the carrier of X and
A27:  xn1 in Ball(0.X,1) and
A28:  s = T.xn1 by A25,FUNCT_2:64;
      reconsider xn1 as Point of X by A26;
A29:  for n being Nat for v,w be Point of X ex z being Point
      of X st P[n,v,w,z]
      proof
        let n being Nat;
        let v,w be Point of X;
        now
          reconsider B0 = Ball(y,e.(n+2)*r) as Subset of LinearTopSpaceNorm Y
          by NORMSP_2:def 4;
          reconsider BYsr = Ball(T.w,e.(n+1)*r) as Subset of
          LinearTopSpaceNorm Y by NORMSP_2:def 4;
          reconsider TB1 = T.: Ball(w,e.(n+1)) as Subset of LinearTopSpaceNorm
          Y by NORMSP_2:def 4;
          assume that
          w in Ball(v,e.n) and
A30:      ||.T.w-y.|| < e.(n+1)*r;
          e.(n+1)=a|^(n+1) by PREPOWER:def 1;
          then
A31:      BYsr c= Cl (TB1) by A5,A23,NEWTON:83;
          e.(n+2)=a|^(n+2) & 0<a|^(n+2) by A23,NEWTON:83,PREPOWER:def 1;
          then 0< e.(n+2)*r by A1,XREAL_1:129;
          then ||.y-y.|| < e.(n+2)*r by NORMSP_1:6;
          then
A32:      B0 is open & y in B0 by NORMSP_2:23;
          y in BYsr by A30;
          then Ball(y,e.(n+2)*r) meets TB1 by A31,A32,PRE_TOPC:def 7;
          then consider s be object such that
A33:      s in Ball(y,e.(n+2)*r) and
A34:      s in T.: Ball(w,e.(n+1)) by XBOOLE_0:3;
          consider z be object such that
A35:      z in the carrier of X and
A36:      z in Ball(w,e.(n+1)) and
A37:      s = T.z by A34,FUNCT_2:64;
          reconsider z as Point of X by A35;
          reconsider sb=T.z as Point of Y;
          ex ss1 be Point of Y st sb=ss1 & ||.y-ss1.|| < e.(n+2)*r by A33,A37;
          then ||.T.z-y.|| < e.(n+2)*r by NORMSP_1:7;
          hence
          ex z be Point of X st z in Ball(w,e.(n+1)) & ||.T.z-y.|| < e.(n
          +2)*r by A36;
        end;
        hence thesis;
      end;
      consider xn be sequence of X such that
A38:  xn.0 = 0.X & xn.1 = xn1 & for n being Nat holds P[n,
      xn.n,xn.(n+1),xn.(n+2)] from RecExD3(A29);
      reconsider sb=T.xn1 as Point of Y;
A39:  ex ss1 be Point of Y st sb=ss1 & ||.y-ss1.|| < e.1*r by A24,A28;
A40:  for n be Nat holds xn.(n+1) in Ball(xn.n,e.n) & ||.T.(xn
      .(n+1))-y.|| < e.(n+1)*r
      proof
        defpred PN[Nat] means xn.($1+1) in Ball(xn.$1,e.$1) & ||.T.
        (xn.($1+1))-y.|| < e.($1+1)*r;
A41:    now
          let n be Nat;
          assume
A42:      PN[n];
          P[n,xn.n,xn.(n+1),xn.(n+2)] by A38;
          hence PN[n+1] by A42;
        end;
A43:    PN[0] by A27,A39,A38,NORMSP_1:7,PREPOWER:3;
        thus for n be Nat holds PN[n] from NAT_1:sch 2(A43, A41);
      end;
A44:  e.0 = 1 by PREPOWER:3;
A45:  for m, k be Nat holds ||.xn.(m+k) - xn.m.|| <= e.m*((1-e
      .k)/(1-e.1))
      proof
        let m be Nat;
        defpred PN[Nat] means ||.xn.(m+$1) - xn.m.|| <= e.m*((1-e.
        $1)/(1-e.1));
A46:    now
          let k be Nat;
A47:      (a|^k-a|^(k+1))/(1-a|^1)+ (1-a|^k)/(1-a|^1) =(a|^k-a|^(k+1)+(1-a
          |^k))/(1-a|^1) by XCMPLX_1:62
            .=(1-a|^(k+1))/(1-a|^1);
          assume PN[k];
          then ||.xn.((m+k)+1) - xn.m.|| <= ||.xn.((m+k)+1) - xn.(m+k).|| +
||.xn.(m+k) - xn .m.|| & ||.xn.((m+k)+1) - xn.(m+k).|| + ||.xn.(m+k) - xn.m.||
<= ||.xn.((m+k)+ 1) - xn.(m+k).|| + e.m*((1-e.k)/(1-e.1)) by NORMSP_1:10
,XREAL_1:6;
          then
A48:      ||.xn.(m+(k+1)) - xn.m.|| <= ||.xn.((m+k)+1) - xn.(m+k).|| + e.
          m*((1-e.k)/(1-e.1)) by XXREAL_0:2;
          xn.(m+k+1) in Ball(xn.(m+k),e.(m+k)) by A40;
          then
          ex xn2 be Point of X st xn.((m+k)+1)=xn2 & ||.xn.(m+k) - xn2.||
          < e.(m+k);
          then ||.xn.((m+k)+1) - xn.(m+k).|| < e.(m+k) by NORMSP_1:7;
          then
A49:      ||.xn.((m+k)+1) - xn.(m+k).|| + e.m*((1-e.k)/(1-e.1)) <= e.(m+k
          ) + e.m*((1-e.k)/(1-e.1)) by XREAL_1:6;
          a|^1<1 by A20;
          then 0< 1-a|^1 by XREAL_1:50;
          then
A50:      a|^k=((a|^k)*(1-a|^1))/(1-a|^1) by XCMPLX_1:89
            .=(a|^k-a|^k*a|^1)/(1-a|^1)
            .=(a|^k-a|^(k+1))/(1-a|^1) by NEWTON:8;
          e.(m+k)+e.m*((1-e.k)/(1-e.1)) = a|^(m+k)+e.m*((1-e.k)/(1-e.1))
          by PREPOWER:def 1
            .= a|^(m+k)+(a|^m)*((1-e.k)/(1-e.1)) by PREPOWER:def 1
            .= a|^(m+k)+(a|^m)*((1-a|^k)/(1-e.1)) by PREPOWER:def 1
            .= a|^(m+k)+(a|^m)*((1-a|^k)/(1-a|^1)) by PREPOWER:def 1
            .=a|^m*a|^k + (a|^m)*((1-a|^k)/(1-a|^1)) by NEWTON:8
            .=a|^m*((1-a|^(k+1))/(1-a|^1)) by A50,A47
            .=e.m*((1-a|^(k+1))/(1-a|^1)) by PREPOWER:def 1
            .=e.m*((1-e.(k+1))/(1-a|^1)) by PREPOWER:def 1
            .=e.m*((1-e.(k+1))/(1-e.1)) by PREPOWER:def 1;
          hence PN[k+1] by A48,A49,XXREAL_0:2;
        end;
        ||.xn.(m+0) - xn.m.|| = ||.0.X.|| by RLVECT_1:5;
        then
A51:    PN[ 0] by A44;
        for k be Nat holds PN[ k] from NAT_1:sch 2(A51,A46);
        hence thesis;
      end;
A52:  for k be Nat st 0 <= k holds ||.xn.||.k <= 1/(1-a)
      proof
        let k be Nat;
        assume 0 <= k;
A53:    e.k = a|^k & e.1 = a|^1 by PREPOWER:def 1;
        a|^1 < 1 by A20;
        then
A54:    0< 1-a|^1 by XREAL_1:50;
        1-a|^k <1 by A23,NEWTON:83,XREAL_1:44;
        then (1-e.k)/(1-e.1) <= 1/(1-a|^1) by A53,A54,XREAL_1:74;
        then
A55:    (1-e.k)/(1-e.1) <= 1/(1-a );
        ||. xn.(0+k) - xn.0 .|| <= e.0*((1-e.k)/(1-e.1)) by A45;
        then ||.xn.k .|| <= (1-e.k)/(1-e.1) by A44,A38,RLVECT_1:13;
        then ||.xn.k .|| <= 1/(1-a ) by A55,XXREAL_0:2;
        hence thesis by NORMSP_0:def 4;
      end;
A56:  Sum(e) = 1/(1-a) by A21,SERIES_1:24;
      1/(1-a)=1/((1*(1+s1)-s1)/(1+s1)) by A18,XCMPLX_1:127
        .=1+s1 by XCMPLX_1:100;
      then
A57:  Sum(e) < 1+s0 by A19,A56,XREAL_1:6;
      set xx= lim xn;
A58:  now
        let m be Nat;
        hereby
          let k be Nat;
A59:      0 < a|^m by A23,NEWTON:83;
          ||.xn.(m+k)-xn.m.|| <= e.m*((1-e.k)/(1-e.1)) by A45;
          then ||.xn.(m+k)-xn.m.|| <= a|^m*((1-e.k)/(1-e.1)) by PREPOWER:def 1;
          then ||.xn.(m+k)-xn.m.|| <= a|^m*((1-a|^k)/(1-e.1)) by PREPOWER:def 1
;
          then
A60:      ||.xn.(m+k)-xn.m.|| <= a|^m*((1-a|^k)/(1-a|^1)) by PREPOWER:def 1;
          a|^1 < 1 by A20;
          then
A61:      0< 1-a|^1 by XREAL_1:50;
          1-a|^k < 1 by A23,NEWTON:83,XREAL_1:44;
          then (1-a|^k)/(1-a|^1)< 1/(1-a|^1) by A61,XREAL_1:74;
          then a|^m*((1-a|^k)/(1-a|^1)) < a|^m*(1/(1-a|^1)) by A59,XREAL_1:68;
          hence ||.xn.(m+k)-xn.m.|| < a|^m*(1/(1-a|^1)) by A60,XXREAL_0:2;
        end;
      end;
      now
        let r1 be Real;
A62:    e is convergent & lim e =0 by A22,SERIES_1:4;
        a|^1 < 1 by A20;
        then
A63:    0< 1-a|^1 by XREAL_1:50;
        assume 0 < r1;
        then 0< r1*(1-a|^1) by A63,XREAL_1:129;
        then consider p1 be Nat such that
A64:    for m be Nat st p1<=m holds |.e.m-0 .|< r1*(1-a|^
        1) by A62,SEQ_2:def 7;
         reconsider p1 as Nat;
        take m=p1;
        |.e.p1-0 .|< r1*(1-a|^1) by A64;
        then e.p1 < 0+r1*(1-a|^1) by RINFSUP1:1;
        then e.p1/(1-a|^1) <r1*(1-a|^1)/(1-a|^1) by A63,XREAL_1:74;
        then e.p1/(1-a|^1) < r1 by A63,XCMPLX_1:89;
        then a|^p1/(1-a|^1) < r1 by PREPOWER:def 1;
        then
A65:    a|^p1*(1/(1-a|^1)) < r1 by XCMPLX_1:99;
        hereby
          let n be Nat;
          assume m <= n;
          then consider k1 be Nat such that
A66:      n = m + k1 by NAT_1:10;
          reconsider k1 as Nat;
          n = m + k1 by A66;
          hence ||.xn.n-xn.m.|| < r1 by A58,A65,XXREAL_0:2;
        end;
      end;
      then xn is Cauchy_sequence_by_Norm by LOPBAN_3:5;
      then
A67:  xn is convergent by LOPBAN_1:def 15;
      then lim ||.xn.|| = ||.lim xn.|| by LOPBAN_1:41;
      then ||.xx .|| <= Sum(e) by A56,A67,A52,LOPBAN_1:41,RSSPACE2:5;
      then ||.xx.|| < 1+s0 by A57,XXREAL_0:2;
      then ||.-xx.|| < 1+s0 by NORMSP_1:2;
      then ||.0.X-xx.|| < 1+s0 by RLVECT_1:14;
      then
A68:  xx in Ball(0.X,1+s0);
      rng xn c= the carrier of X;
      then
A69:  rng xn c= dom T by FUNCT_2:def 1;
A70:  now
        let n be Nat;
A71:  n in NAT by ORDINAL1:def 12;
        thus (T/*xn).n=T/.(xn.n) by A69,FUNCT_2:109,A71
          .=T.(xn.n);
      end;
A72:  now
        let s be Real;
        assume 0 < s;
        then
A73:    0< s/r by A1,XREAL_1:139;
        e is convergent & lim e =0 by A22,SERIES_1:4;
        then consider m0 be Nat such that
A74:    for n be Nat st m0<=n holds |.e.n-0 .|< s/r by A73,
SEQ_2:def 7;
         reconsider m=m0+1 as Nat;
        take m;
        a|^1 < 1 & 0 < a|^m0 by A23,A20,NEWTON:83;
        then a|^m0*a|^1 <= a|^m0 by XREAL_1:153;
        then a|^(m0+1) <= a|^m0 by NEWTON:8;
        then e.(m0+1) <= a|^m0 by PREPOWER:def 1;
        then
A75:    e.(m0+1) <= e.m0 by PREPOWER:def 1;
        |.e.m0-0 .|< s/r by A74;
        then e.m0 < 0+s/r by RINFSUP1:1;
        then e.(m0+1)< s/r by A75,XXREAL_0:2;
        then e.(m0+1)*r < s/r*r by A1,XREAL_1:68;
        then
A76:    e.(m0+1)*r < s by A1,XCMPLX_1:87;
        now
          let n be Nat;
          assume
A77:      m <= n;
          1 <= m0+1 by NAT_1:11;
          then reconsider n0= n-1 as Nat by A77,NAT_1:21,XXREAL_0:2;
          consider m1 be Nat such that
A78:      n0+1 = m0+1 + m1 by A77,NAT_1:10;
A79:      a |^(n0+1) = a |^(m0+1) * a |^ m1 by A78,NEWTON:8;
          0< a |^(m0+1) & a|^m1 <= 1|^m1 by A23,A20,NEWTON:83,PREPOWER:9;
          then a |^(n0+1) <= a |^(m0+1) by A79,XREAL_1:153;
          then e.(n0+1) <= a |^(m0+1) by PREPOWER:def 1;
          then e.(n0+1) <= e.(m0+1) by PREPOWER:def 1;
          then e.(n0+1)*r <= e.(m0+1)*r by A1,XREAL_1:64;
          then ||.T.(xn.n)-y.|| <= e.(m0+1)*r by A40,XXREAL_0:2;
          then ||.T.(xn.n)-y.|| < s by A76,XXREAL_0:2;
          hence ||.(T/*xn).n-y.|| < s by A70;
        end;
        hence for n be Nat st m <= n holds ||.(T/*xn).n-y.|| < s;
      end;
      T is_continuous_in xx by LOPBAN_5:4;
      then T/*xn is convergent & T/.xx = lim (T/*xn) by A67,A69,NFCONT_1:def 5;
      then y=T.xx by A72,NORMSP_1:def 7;
      hence z in T.:Ball(0.X,1+s0) by A68,FUNCT_2:35;
    end;
    hence thesis;
  end;
  now
    reconsider TB01 = T.: Ball(0.X,1) as Subset of Y;
    let z be object;
    assume
A80: z in Ball(0.Y,r);
    then reconsider y=z as Point of Y;
    ex yy1 be Point of Y st y=yy1 & ||.0.Y-yy1.|| < r by A80;
    then ||.-y.|| < r by RLVECT_1:14;
    then
A81: ||.y.|| < r by NORMSP_1:2;
    consider s0 be Real such that
A82: 0<s0 and
A83: ||.y.||< r/(1+s0) and
    r/(1+s0) < r by A81,Th1;
    set y1=(1+s0)*y;
    (1+s0)*||.y.|| < r/(1+s0)*(1+s0) by A82,A83,XREAL_1:68;
    then (1+s0)*||.y.|| < r by A82,XCMPLX_1:87;
    then |.1+s0.|*||.y.|| < r by A82,ABSVALUE:def 1;
    then ||.(1+s0)*y.|| < r by NORMSP_1:def 1;
    then ||.-y1.|| < r by NORMSP_1:2;
    then ||.0.Y-y1.|| < r by RLVECT_1:14;
    then
A84: (1+s0)*y in Ball(0.Y,r);
    Ball(0.Y,r) c= T.: Ball(0.X,1+s0) by A15,A82;
    then
A85: (1+s0)*y in T.: Ball(0.X,1+s0) by A84;
    reconsider s1=1+s0 as non zero Real by A82;
    s1*Ball(0.X,1)= Ball(0.X,s1*1) by A82,Th3;
    then s1*y in s1*TB01 by A85,Th5;
    then s1"*(s1*y) in s1"*(s1*TB01);
    then (s1"*s1)*y in s1"*(s1*TB01) by RLVECT_1:def 7;
    then
A86: (s1"*s1)*y in (s1"*s1)*TB01 by CONVEX1:37;
    s1"*s1=(1/s1)*s1 by XCMPLX_1:215
      .=1 by XCMPLX_1:106;
    then y in 1*TB01 by A86,RLVECT_1:def 8;
    hence z in T.: Ball(0.X,1) by CONVEX1:32;
  end;
  hence thesis by A2,A3;
end;
