reserve X, Y for RealNormSpace;

theorem
  for X, Y be RealBanachSpace, T be Lipschitzian LinearOperator of X,Y, T1 be
  Function of LinearTopSpaceNorm X,LinearTopSpaceNorm Y st T1=T & T1 is onto
  holds T1 is open
proof
  let X, Y be RealBanachSpace, T be Lipschitzian LinearOperator of X,Y, T1 be
  Function of LinearTopSpaceNorm X,LinearTopSpaceNorm Y;
  assume that
A1: T1=T and
A2: T1 is onto;
  thus for G be Subset of LinearTopSpaceNorm X st G is open holds T1.:G is open
  proof
    reconsider TB1 = T.: Ball(0.X,1) as Subset of LinearTopSpaceNorm Y by
NORMSP_2:def 4;
    defpred P[Nat, set] means ex TBn be Subset of TopSpaceNorm Y st
    TBn=T.: Ball(0.X,$1) & $2=Cl(TBn);
    let G be Subset of LinearTopSpaceNorm X;
A3: for n being Element of NAT ex y being Element of bool the carrier of Y
    st P[n,y]
    proof
      let n being Element of NAT;
      reconsider TBn = T.: Ball(0.X,n) as Subset of TopSpaceNorm Y;
      Cl(TBn) c= the carrier of Y;
      hence thesis;
    end;
    consider f be sequence of bool (the carrier of Y) such that
A4: for x being Element of NAT holds P[x,f.x] from FUNCT_2:sch 3(A3);
    reconsider f as SetSequence of Y;
A5: for n be Nat holds f.n is closed
    proof
      let n be Nat;
      n in NAT by ORDINAL1:def 12;
      then
      ex TBn be Subset of TopSpaceNorm Y st TBn=T.: Ball(0.X,n) & f.n=Cl(
      TBn) by A4;
      hence thesis by NORMSP_2:15;
    end;
A6: the carrier of Y c= union rng f
    proof
      let z be object;
      assume z in the carrier of Y;
      then reconsider z1=z as Point of Y;
      rng T = the carrier of (LinearTopSpaceNorm Y ) by A1,A2,FUNCT_2:def 3;
      then rng T = the carrier of Y by NORMSP_2:def 4;
      then consider x1 be object such that
A7:   x1 in the carrier of X and
A8:   z1=T.x1 by FUNCT_2:11;
      reconsider x1 as Point of X by A7;
      consider m be Element of NAT such that
A9:   ||.x1.|| <= m by MESFUNC1:8;
      set n=m+1;
      ||.x1.|| +0< m+1 by A9,XREAL_1:8;
      then ||.-x1.|| < n by NORMSP_1:2;
      then ||.0.X-x1.|| < n by RLVECT_1:14;
      then x1 in Ball(0.X,n);
      then
A10:  T.x1 in T.:Ball(0.X,n) by FUNCT_2:35;
      NAT=dom f by FUNCT_2:def 1;
      then
A11:  f.n in rng f by FUNCT_1:3;
      consider TBn be Subset of TopSpaceNorm Y such that
A12:  TBn=T.: Ball(0.X,n) and
A13:  f.n =Cl(TBn) by A4;
      TBn c= f.n by A13,PRE_TOPC:18;
      hence thesis by A8,A10,A12,A11,TARSKI:def 4;
    end;
    union rng f is Subset of Y by PROB_1:11;
    then union rng f = the carrier of Y by A6;
    then consider n0 be Nat, r be Real,
       y0 be Point of Y such that
A14: 0 < r and
A15: Ball (y0,r) c= f.n0 by A5,LOPBAN_5:3;
    n0 in NAT by ORDINAL1:def 12;
    then consider TBn0 be Subset of TopSpaceNorm Y such that
A16: TBn0=T.: Ball(0.X,n0) and
A17: f.n0=Cl(TBn0) by A4;
    consider TBn01 be Subset of TopSpaceNorm Y such that
A18: TBn01=T.: Ball(0.X,n0+1) and
A19: f.(n0+1)=Cl(TBn01) by A4;
    Ball(0.X,n0) c= Ball(0.X,n0+1) by Th14,NAT_1:11;
    then TBn0 c= TBn01 by A16,A18,RELAT_1:123;
    then f.n0 c= f.(n0+1) by A17,A19,PRE_TOPC:19;
    then
A20: Ball (y0,r) c= Cl(TBn01) by A15,A19;
    reconsider LTBn01=TBn01 as Subset of LinearTopSpaceNorm(Y) by
NORMSP_2:def 4;
    (-1) is non zero Real;
    then
A21: Cl((-1)*LTBn01) =(-1)*Cl(LTBn01) by RLTOPSP1:52;
    reconsider yy0 = y0 as Point of LinearTopSpaceNorm Y by NORMSP_2:def 4;
A22: Ball (0.Y,r/(2*n0+2)) is Subset of LinearTopSpaceNorm Y by NORMSP_2:def 4;
    ||.y0-y0.|| < r by A14,NORMSP_1:6;
    then y0 in Ball (y0,r);
    then y0 in Cl(TBn01) by A20;
    then yy0 in Cl(LTBn01) by Th10;
    then
A23: (-1)*yy0 in (-1)*Cl(LTBn01);
    reconsider nb1 =1/(2*n0+2) as non zero Real by XREAL_1:139;
    reconsider TBX1 = T.: Ball(0.X,1) as Subset of Y;
    reconsider my0 = {-yy0} as Subset of LinearTopSpaceNorm Y;
    reconsider TBnx01 = T.: Ball(0.X,n0+1) as Subset of LinearTopSpaceNorm Y
    by NORMSP_2:def 4;
    reconsider BYyr = Ball(y0,r) as Subset of LinearTopSpaceNorm Y by
NORMSP_2:def 4;
    reconsider BYr = Ball(0.Y,r) as Subset of LinearTopSpaceNorm Y by
NORMSP_2:def 4;
    reconsider XTB01 = T.: Ball(0.X,n0+1) as Subset of Y;
A24: -yy0=(-1)*yy0 by RLVECT_1:16
      .=(-1)*y0 by NORMSP_2:def 4
      .=-y0 by RLVECT_1:16;
    (-1)*LTBn01 = (-1)* (XTB01) by A18,Th9
      .= T.:((-1)*Ball(0.X,n0+1)) by Th5
      .= LTBn01 by A18,Th11;
    then -yy0 in Cl(LTBn01) by A21,A23,RLVECT_1:16;
    then -yy0 in Cl(TBn01) by Th10;
    then {-yy0} c= Cl(TBn01) by ZFMISC_1:31;
    then
A25: my0 c= Cl(TBnx01) by A18,Th10;
    BYyr c= Cl(TBnx01) by A18,A20,Th10;
    then my0 + BYyr c= Cl(TBnx01) + Cl(TBnx01) by A25,RLTOPSP1:11;
    then -yy0 + BYyr c= Cl(TBnx01) + Cl(TBnx01) by RUSUB_4:33;
    then
A26: -y0 + Ball (y0,r) c= Cl(TBnx01) + Cl(TBnx01) by A24,Th8;
A27: Cl(TBnx01) + Cl(TBnx01) c= Cl(TBnx01+TBnx01) by RLTOPSP1:62;
    Ball (y0,r) = y0 + Ball (0.Y,r) by Th2;
    then BYyr = yy0+ BYr by Th8;
    then (-yy0)+BYyr =(-yy0 + yy0)+ BYr by RLTOPSP1:6;
    then (-yy0)+BYyr =0.(LinearTopSpaceNorm Y)+ BYr by RLVECT_1:5;
    then (-yy0)+BYyr ={0.(LinearTopSpaceNorm Y)}+ BYr by RUSUB_4:33;
    then (-yy0)+BYyr = BYr by CONVEX1:35;
    then Ball (0.Y,r) = -y0 + Ball (y0,r) by A24,Th8;
    then
A28: Ball (0.Y,r) c= Cl(TBnx01+TBnx01) by A26,A27;
    TBnx01= 1* TBnx01 by CONVEX1:32;
    then
A29: TBnx01+TBnx01 =(1+1)* TBnx01 by Th13,CONVEX1:13
      .= 2 * TBnx01;
    Ball(0.X,(n0+1)*1) = (n0+1) * Ball(0.X,1) by Th3;
    then (n0+1)*TBX1 = T.: Ball(0.X,n0+1) by Th5;
    then TBnx01+TBnx01 = 2* ((n0+1)*TB1) by A29,Th9
      .= (2*(n0+1))*TB1 by CONVEX1:37
      .=(2*n0+2)*TB1;
    then
A30: Cl(TBnx01+TBnx01) = (2*n0+2)* Cl(TB1) by RLTOPSP1:52;
A31: 0 < r/(2*n0+2) by A14,XREAL_1:139;
    Ball (0.Y, r/(2*n0+2)) = Ball (0.Y, r*1/(2*n0+2))
      .=Ball (0.Y, r*(1/(2*n0+2))) by XCMPLX_1:74
      .=nb1 * Ball(0.Y,r) by Th3
      .= nb1 * BYr by Th9;
    then
A32: Ball (0.Y, r/(2*n0+2)) c= 1/(2*n0+2) *((2*n0+2)* Cl(TB1)) by A28,A30,
CONVEX1:39;
    1/(2*n0+2) *((2*n0+2)* Cl(TB1)) =(1/(2*n0+2)*(2*n0+2))* Cl(TB1) by
CONVEX1:37
      .= 1* Cl(TB1) by XCMPLX_1:106
      .= Cl(TB1) by CONVEX1:32;
    then
A33: Ball (0.Y,r/(2*n0+2)) c= T.: Ball(0.X,1) by A14,A32,A22,Th15,XREAL_1:139;
A34: for p be Real st p > 0
    ex q be Real st 0 < q & Ball (0.Y,q) c=T.:
    Ball(0.X,p)
    proof
      reconsider TB1 =T.: Ball(0.X,1) as Subset of Y;
      let p be Real;
      assume
A35:  p > 0;
      then
A36:  p*Ball(0.X,1)= Ball(0.X,p*1) by Th3;
      take r/(2*n0+2)*p;
      p* Ball (0.Y,r/(2*n0+2)) c= p*TB1 by A33,CONVEX1:39;
      then Ball (0.Y,r/(2*n0+2)*p) c= p*TB1 by A35,Th3;
      hence thesis by A31,A35,A36,Th5,XREAL_1:129;
    end;
    assume
A37: G is open;
    now
      let y be Point of Y;
      assume y in T1.:G;
      then consider x be Point of X such that
A38:  x in G and
A39:  y=T.x by A1,FUNCT_2:65;
      consider p be Real such that
A40:  p>0 and
A41:  {z where z is Point of X: ||.x-z.|| < p} c= G by A37,A38,NORMSP_2:22;
      reconsider TBp =T.: Ball(0.X,p) as Subset of Y;
      consider q be Real such that
A42:  0 < q and
A43:  Ball (0.Y,q) c=TBp by A34,A40;
      Ball(x,p) c= G by A41;
      then
A44:  x+Ball(0.X,p) c= G by Th2;
      now
        let t be object;
        assume t in y + TBp;
        then consider tz0 be Point of Y such that
A45:    t=y+tz0 and
A46:    tz0 in TBp;
        consider z0 be Element of X such that
A47:    z0 in Ball(0.X,p) and
A48:    tz0=T.z0 by A46,FUNCT_2:65;
        reconsider z0 as Point of X;
A49:    x+ z0 in x+Ball(0.X,p) by A47;
        t=T.(x+z0) by A39,A45,A48,VECTSP_1:def 20;
        hence t in T1.:G by A1,A44,A49,FUNCT_2:35;
      end;
      then
A50:  y + TBp c= T.: G by A1;
      take q;
      now
        let t be object;
        assume t in y + Ball(0.Y,q);
        then ex z0 be Point of Y st t= y + z0 & z0 in Ball(0.Y,q);
        hence t in y + TBp by A43;
      end;
      then y + Ball (0.Y,q) c= y+ TBp;
      then Ball (y,q) c= y+ TBp by Th2;
      hence 0 < q & {w where w is Point of Y: ||.y-w.|| < q } c= T1.:G by A1
,A50,A42;
    end;
    hence thesis by NORMSP_2:22;
  end;
end;
