reserve T,T1,T2,S for non empty TopSpace;
reserve p,q for Point of TOP-REAL 2;

theorem Th42:
  for f,g being Function of I[01],TOP-REAL 2, K0 being Subset of
TOP-REAL 2, O,I being Point of I[01] st O=0 & I=1 & f is continuous one-to-one
& g is continuous one-to-one & K0={p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1}& (f.O)`1
=-1 & (f.I)`1=1 & -1<=(f.O)`2 & (f.O)`2<=1 & -1<=(f.I)`2 & (f.I)`2<=1 & (g.O)`2
  =-1 & (g.I)`2=1 & -1<=(g.O)`1 & (g.O)`1<=1 & -1<=(g.I)`1 & (g.I)`1<=1 & rng f
  misses K0 & rng g misses K0 holds rng f meets rng g
proof
  reconsider B={0.TOP-REAL 2} as Subset of TOP-REAL 2;
A1: B`<>{} by Th9;
  reconsider W=B` as non empty Subset of TOP-REAL 2 by Th9;
  defpred P[Point of TOP-REAL 2] means -1=$1`1 & -1<=$1`2 & $1`2<=1 or $1`1=1
& -1<=$1`2 & $1`2<=1 or -1=$1`2 & -1<=$1`1 & $1`1<=1 or 1=$1`2 & -1<=$1`1 & $1
  `1<=1;
A2: the carrier of (TOP-REAL 2)|B` =[#]((TOP-REAL 2)|B`)
    .=B` by PRE_TOPC:def 5;
  reconsider Kb={q: P[q]} as Subset of TOP-REAL 2 from TopSubset;
  let f,g be Function of I[01],TOP-REAL 2, K0 be Subset of TOP-REAL 2, O,I be
  Point of I[01];
A3: dom f =the carrier of I[01] by FUNCT_2:def 1;
  assume
A4: O=0 & I=1 & f is continuous & f is one-to-one & g is continuous & g
is one-to-one & K0={p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1}& (f.O)`1=-1 & (f.I)`1=1
& -1<=(f.O)`2 & (f.O)`2<=1 & -1<=(f.I)`2 & (f.I)`2<=1 & (g.O)`2=-1 & (g.I)`2=1
& -1<=(g.O)`1 & (g.O)`1<=1 & -1<=(g.I)`1 & (g.I)`1<=1 & rng f /\ K0={} & rng g
  /\ K0={};
  then consider h being Function of (TOP-REAL 2)|B`,(TOP-REAL 2)|B` such that
A5: h is continuous and
A6: h is one-to-one and
  for t being Point of TOP-REAL 2 st t in K0 & t<>0.TOP-REAL 2 holds not h
  .t in K0 \/ Kb and
A7: for r being Point of TOP-REAL 2 st not r in K0 \/ Kb holds h.r in K0 and
A8: for s being Point of TOP-REAL 2 st s in Kb holds h.s=s by Th41;
  rng f c= B`
  proof
    let x be object;
    assume
A9: x in rng f;
    now
      assume x in B;
      then
A10:  x=0.TOP-REAL 2 by TARSKI:def 1;
      (0.TOP-REAL 2)`1=0 & (0.TOP-REAL 2)`2=0 by EUCLID:52,54;
      then 0.TOP-REAL 2 in K0 by A4;
      hence contradiction by A4,A9,A10,XBOOLE_0:def 4;
    end;
    then x in (the carrier of TOP-REAL 2)\ B by A9,XBOOLE_0:def 5;
    hence thesis by SUBSET_1:def 4;
  end;
  then
A11: ex w being Function of I[01],TOP-REAL 2 st w is continuous & w=h*f by A4
,A5,A1,Th12;
  then reconsider d1=h*f as Function of I[01],TOP-REAL 2;
  the carrier of (TOP-REAL 2)|W <>{};
  then
A12: dom h=the carrier of (TOP-REAL 2)|B` by FUNCT_2:def 1;
  rng g c= B`
  proof
    let x be object;
    assume
A13: x in rng g;
    now
      assume x in B;
      then
A14:  x=0.TOP-REAL 2 by TARSKI:def 1;
      0.TOP-REAL 2 in K0 by A4,Th3;
      hence contradiction by A4,A13,A14,XBOOLE_0:def 4;
    end;
    then x in (the carrier of TOP-REAL 2)\ B by A13,XBOOLE_0:def 5;
    hence thesis by SUBSET_1:def 4;
  end;
  then
A15: ex w2 being Function of I[01],TOP-REAL 2 st w2 is continuous & w2=h*g by
A4,A5,A1,Th12;
  then reconsider d2=h*g as Function of I[01],TOP-REAL 2;
A16: dom g =the carrier of I[01] by FUNCT_2:def 1;
A17: for r being Point of I[01] holds -1<=(d1.r)`1 & (d1.r)`1<=1 & -1<=(d2.r
  )`1 & (d2.r)`1<=1 & -1<=(d1.r)`2 & (d1.r)`2<=1 & -1<=(d2.r)`2 & (d2.r)`2<=1
  proof
    let r be Point of I[01];
A18: g.r in Kb implies d2.r in K0 \/ Kb
    proof
A19:  d2.r=h.(g.r) by A16,FUNCT_1:13;
      assume
A20:  g.r in Kb;
      then h.(g.r)=g.r by A8;
      hence thesis by A20,A19,XBOOLE_0:def 3;
    end;
    f.r in rng f by A3,FUNCT_1:3;
    then
A21: not f.r in K0 by A4,XBOOLE_0:def 4;
A22: not f.r in Kb implies d1.r in K0 \/ Kb
    proof
      assume not f.r in Kb;
      then not f.r in K0 \/ Kb by A21,XBOOLE_0:def 3;
      then
A23:  h.(f.r) in K0 by A7;
      d1.r=h.(f.r) by A3,FUNCT_1:13;
      hence thesis by A23,XBOOLE_0:def 3;
    end;
    g.r in rng g by A16,FUNCT_1:3;
    then
A24: not g.r in K0 by A4,XBOOLE_0:def 4;
A25: not g.r in Kb implies d2.r in K0 \/ Kb
    proof
      assume not g.r in Kb;
      then not g.r in K0 \/ Kb by A24,XBOOLE_0:def 3;
      then
A26:  h.(g.r) in K0 by A7;
      d2.r=h.(g.r) by A16,FUNCT_1:13;
      hence thesis by A26,XBOOLE_0:def 3;
    end;
A27: f.r in Kb implies d1.r in K0 \/ Kb
    proof
A28:  d1.r=h.(f.r) by A3,FUNCT_1:13;
      assume
A29:  f.r in Kb;
      then h.(f.r)=f.r by A8;
      hence thesis by A29,A28,XBOOLE_0:def 3;
    end;
    now
      per cases by A22,A27,A25,A18,XBOOLE_0:def 3;
      case
        d1.r in K0 & d2.r in K0;
        then
        (ex p being Point of TOP-REAL 2 st p=d1.r & -1<p`1 & p`1 <1 & -1<
p`2 & p `2<1 )& ex q being Point of TOP-REAL 2 st q=d2.r & -1<q`1 & q`1 <1 & -1
        <q `2 & q`2<1 by A4;
        hence thesis;
      end;
      case
        d1.r in K0 & d2.r in Kb;
        then
        (ex p being Point of TOP-REAL 2 st p=d1.r & -1<p`1 & p`1 <1 & -1<
p`2 & p `2<1 )& ex q being Point of TOP-REAL 2 st q=d2.r &( -1=q`1 & -1<= q`2 &
q`2<= 1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1
        <=q`1 & q`1<=1) by A4;
        hence thesis;
      end;
      case
        d1.r in Kb & d2.r in K0;
        then
        (ex p being Point of TOP-REAL 2 st p=d2.r & -1<p`1 & p`1 <1 & -1<
p`2 & p `2<1 )& ex q being Point of TOP-REAL 2 st q=d1.r &( -1=q`1 & -1<= q`2 &
q`2<= 1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1
        <=q`1 & q`1<=1) by A4;
        hence thesis;
      end;
      case
        d1.r in Kb & d2.r in Kb;
        then (ex p being Point of TOP-REAL 2 st p=d2.r &( -1=p`1 & -1<= p`2 &
p`2<=1 or p `1=1 & -1<=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or 1=p`2 & -1
<=p`1 & p`1<= 1))& ex q being Point of TOP-REAL 2 st q=d1.r &( -1=q`1 & -1<= q
`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 &
        -1<=q`1 & q `1<=1);
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  f.I in Kb by A4;
  then h.(f.I)=f.I by A8;
  then
A30: (d1.I)`1=1 by A4,A3,FUNCT_1:13;
  f.O in Kb by A4;
  then h.(f.O)=f.O by A8;
  then
A31: (d1.O)`1=-1 by A4,A3,FUNCT_1:13;
  g.I in Kb by A4;
  then h.(g.I)=g.I by A8;
  then
A32: (d2.I)`2=1 by A4,A16,FUNCT_1:13;
  g.O in Kb by A4;
  then h.(g.O)=g.O by A8;
  then
A33: (d2.O)`2=-1 by A4,A16,FUNCT_1:13;
  set s = the Element of rng d1 /\ rng d2;
  d1 is one-to-one & d2 is one-to-one by A4,A6,FUNCT_1:24;
  then rng d1 meets rng d2 by A4,A11,A15,A31,A30,A33,A32,A17,JGRAPH_1:47;
  then
A34: rng d1 /\ rng d2<>{};
  then s in rng d1 by XBOOLE_0:def 4;
  then consider t1 being object such that
A35: t1 in dom d1 and
A36: s=d1.t1 by FUNCT_1:def 3;
A37: f.t1 in rng f by A3,A35,FUNCT_1:3;
  s in rng d2 by A34,XBOOLE_0:def 4;
  then consider t2 being object such that
A38: t2 in dom d2 and
A39: s=d2.t2 by FUNCT_1:def 3;
  h.(f.t1)=d1.t1 by A35,FUNCT_1:12;
  then
A40: h.(f.t1)=h.(g.t2) by A36,A38,A39,FUNCT_1:12;
  rng g c=(the carrier of (TOP-REAL 2))\B
  proof
    let e be object;
    assume
A41: e in rng g;
    now
      assume e in B;
      then
A42:  e=0.TOP-REAL 2 by TARSKI:def 1;
      0.TOP-REAL 2 in {p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1} by Th3;
      hence contradiction by A4,A41,A42,XBOOLE_0:def 4;
    end;
    hence thesis by A41,XBOOLE_0:def 5;
  end;
  then
A43: rng g c= the carrier of (TOP-REAL 2)|B` by A2,SUBSET_1:def 4;
  dom g =the carrier of I[01] by FUNCT_2:def 1;
  then
A44: g.t2 in rng g by A38,FUNCT_1:3;
  rng f c=(the carrier of (TOP-REAL 2))\B
  proof
    let e be object;
    assume
A45: e in rng f;
    now
      assume e in B;
      then
A46:  e=0.TOP-REAL 2 by TARSKI:def 1;
      0.TOP-REAL 2 in {p: -1<p`1 & p`1<1 & -1<p`2 & p`2<1} by Th3;
      hence contradiction by A4,A45,A46,XBOOLE_0:def 4;
    end;
    hence thesis by A45,XBOOLE_0:def 5;
  end;
  then rng f c= the carrier of (TOP-REAL 2)|B` by A2,SUBSET_1:def 4;
  then f.t1=g.t2 by A6,A43,A40,A12,A37,A44,FUNCT_1:def 4;
  then rng f /\ rng g <> {} by A37,A44,XBOOLE_0:def 4;
  hence thesis;
end;
