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

theorem
  for f,g being Function of I[01],TOP-REAL 2,a,b,c,d being Real ,
  O,I being Point of I[01] st O=0 & I=1 & f is continuous one-to-one & g is
continuous one-to-one & (f.O)`1=a & (f.I)`1=b & c <=(f.O)`2 & (f.O)`2<=d & c <=
(f.I)`2 & (f.I)`2<=d & (g.O)`2=c & (g.I)`2=d & a<=(g.O)`1 & (g.O)`1<=b & a<=(g.
I)`1 & (g.I)`1<=b & a < b & c < d & not (ex r being Point of I[01] st a<(f.r)`1
& (f.r)`1<b & c <(f.r)`2 & (f.r)`2<d)& not (ex r being Point of I[01] st a<(g.r
  )`1 & (g.r)`1<b & c <(g.r)`2 & (g.r)`2<d) holds rng f meets rng g
proof
  defpred P[Point of TOP-REAL 2] means -1<$1`1 & $1`1<1 & -1<$1`2 & $1`2<1;
  reconsider K0={p: P[p]} as Subset of TOP-REAL 2 from TopSubset;
  let f,g be Function of I[01],TOP-REAL 2,a,b,c,d be Real, O,I be Point
  of I[01];
  assume that
A1: O=0 & I=1 and
A2: f is continuous one-to-one & g is continuous one-to-one and
A3: (f.O)`1=a and
A4: (f.I)`1=b and
A5: c <=(f.O)`2 and
A6: (f.O)`2<=d and
A7: c <=(f.I)`2 and
A8: (f.I)`2<=d and
A9: (g.O)`2=c and
A10: (g.I)`2=d and
A11: a<=(g.O)`1 and
A12: (g.O)`1<=b and
A13: a<=(g.I)`1 and
A14: (g.I)`1<=b and
A15: a < b and
A16: c < d and
A17: not(ex r being Point of I[01] st a<(f.r)`1 & (f.r)`1<b & c <(f.r)`2
  & (f.r)`2<d) and
A18: not(ex r being Point of I[01] st a<(g.r)`1 & (g.r)`1<b & c <(g.r)`2
  & (g.r)`2<d);
  set A=2/(b-a),B=1-2*b/(b-a),C=2/(d-c),D=1-2*d/(d-c);
  set ff =AffineMap(A,B,C,D);
  reconsider f2=ff*f,g2=ff*g as Function of I[01],TOP-REAL 2;
A19: d-c>0 by A16,XREAL_1:50;
  then
A20: C>0 by XREAL_1:139;
A21: dom g=the carrier of I[01] by FUNCT_2:def 1;
  then
A22: g2.I=ff.(g.I) by FUNCT_1:13
    .=|[A*((g.I)`1)+B,C*d+D]| by A10,Def2;
  then
A23: (g2.I)`2=C*d+D by EUCLID:52
    .= d*2/(d-c)+(1-2*d/(d-c))
    .= 1;
A24: g2.O=ff.(g.O) by A21,FUNCT_1:13
    .=|[A*((g.O)`1)+B,C*c+D]| by A9,Def2;
  then
A25: (g2.O)`2=2/(d-c)*c+(1-2*d/(d-c)) by EUCLID:52
    .= c*2/(d-c)+(1-2*d/(d-c))
    .= c*2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A19,XCMPLX_1:60
    .= c*2/(d-c)+((d-c)-2*d)/(d-c)
    .= (c*2+((d-c)-2*d))/(d-c)
    .= (-(d-c))/(d-c)
    .= -((d-c)/(d-c))
    .= -1 by A19,XCMPLX_1:60;
A26: b-a>0 by A15,XREAL_1:50;
A27: -1<=(g2.O)`1 & (g2.O)`1<=1 & -1<=(g2.I)`1 & (g2.I)`1<=1
  proof
    reconsider s1=(g.I)`1 as Real;
    reconsider s0=(g.O)`1 as Real;
A28: (a-b)/(b-a) = (-(b-a))/(b-a) .= -((b-a)/(b-a))
      .= -1 by A26,XCMPLX_1:60;
A29: (g2.I)`1=A*s1+B by A22,EUCLID:52
      .= s1*2/(b-a)+(1-2*b/(b-a))
      .= s1*2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A26,XCMPLX_1:60
      .= s1*2/(b-a)+((b-a)-2*b)/(b-a)
      .= (s1*2+((b-a)-2*b))/(b-a)
      .= ((s1-b)+(s1-b)-(a-b))/(b-a);
    b-b>=s0-b by A12,XREAL_1:9;
    then 0+(b-b)-(a-b)>=(s0-b)+(s0-b)-(a-b) by XREAL_1:9;
    then
A30: (b-a)/(b-a)>=((s0-b)+(s0-b)-(a-b))/(b-a) by A26,XREAL_1:72;
    b-b>=s1-b by A14,XREAL_1:9;
    then
A31: 0+(b-b)-(a-b)>=(s1-b)+(s1-b)-(a-b) by XREAL_1:9;
    a-b<=s1-b by A13,XREAL_1:9;
    then a-b+(a-b)<=(s1-b)+(s1-b) by XREAL_1:7;
    then
A32: a-b+(a-b)-(a-b)<=(s1-b)+(s1-b)-(a-b) by XREAL_1:9;
    a-b<=s0-b by A11,XREAL_1:9;
    then a-b+(a-b)<=(s0-b)+(s0-b) by XREAL_1:7;
    then
A33: a-b+(a-b)-(a-b)<=(s0-b)+(s0-b)-(a-b) by XREAL_1:9;
    (g2.O)`1=A*s0+B by A24,EUCLID:52
      .= s0 *2/(b-a)+(1-2*b/(b-a))
      .= s0 *2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A26,XCMPLX_1:60
      .= s0 *2/(b-a)+((b-a)-2*b)/(b-a)
      .= (s0 *2+((b-a)-2*b))/(b-a)
      .= ((s0-b)+(s0-b)-(a-b))/(b-a);
    hence thesis by A26,A33,A28,A30,A29,A32,A31,XREAL_1:72;
  end;
A34: now
    assume rng f2 meets K0;
    then consider x being object such that
A35: x in rng f2 and
A36: x in K0 by XBOOLE_0:3;
    reconsider q=x as Point of TOP-REAL 2 by A35;
    consider p such that
A37: p=q and
A38: -1<p`1 and
A39: p`1<1 and
A40: -1<p`2 and
A41: p`2<1 by A36;
    consider z being object such that
A42: z in dom f2 and
A43: x=f2.z by A35,FUNCT_1:def 3;
    reconsider u=z as Point of I[01] by A42;
    reconsider t=f.u as Point of TOP-REAL 2;
A44: A*(t`1)+B= (t`1)*2/(b-a)+(1-2*b/(b-a))
      .= (t`1)*2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A26,XCMPLX_1:60
      .= (t`1)*2/(b-a)+((b-a)-2*b)/(b-a)
      .= ((t`1)*2+((b-a)-2*b))/(b-a)
      .= (2*((t`1)-b)-(a-b))/(b-a);
A45: ff.t=p by A37,A42,A43,FUNCT_1:12;
A46: C*(t`2)+D= (t`2)*2/(d-c)+(1-2*d/(d-c))
      .= (t`2)*2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A19,XCMPLX_1:60
      .= (t`2)*2/(d-c)+((d-c)-2*d)/(d-c)
      .= ((t`2)*2+((d-c)-2*d))/(d-c)
      .= (2*((t`2)-d)-(c-d))/(d-c);
A47: ff.t=|[A*(t`1)+B,C*(t`2)+D]| by Def2;
    then -1<C*(t`2)+D by A40,A45,EUCLID:52;
    then (-1)*(d-c)< (2*((t`2)-d)-(c-d))/(d-c)*(d-c) by A19,A46,XREAL_1:68;
    then (-1)*(d-c)< 2*((t`2)-d)-(c-d) by A19,XCMPLX_1:87;
    then (-1)*(d-c)+(c-d)< 2*((t`2)-d)-(c-d)+(c-d) by XREAL_1:8;
    then 2*(c-d)/2< 2*((t`2)-d)/2 by XREAL_1:74;
    then
A48: c < (t`2) by XREAL_1:9;
    C*(t`2)+D<1 by A41,A47,A45,EUCLID:52;
    then (1)*(d-c)> (2*((t`2)-d)-(c-d))/(d-c)*(d-c) by A19,A46,XREAL_1:68;
    then (1)*(d-c)> 2*((t`2)-d)-(c-d) by A19,XCMPLX_1:87;
    then (1)*(d-c)+(c-d)> 2*((t`2)-d)-(c-d)+(c-d) by XREAL_1:8;
    then 0/2>((t`2)-d)*2/2;
    then
A49: 0+d>t`2 by XREAL_1:19;
    A*(t`1)+B<1 by A39,A47,A45,EUCLID:52;
    then (1)*(b-a)> (2*((t`1)-b)-(a-b))/(b-a)*(b-a) by A26,A44,XREAL_1:68;
    then (1)*(b-a)> 2*((t`1)-b)-(a-b) by A26,XCMPLX_1:87;
    then (1)*(b-a)+(a-b)> 2*((t`1)-b)-(a-b)+(a-b) by XREAL_1:8;
    then 0/2>((t`1)-b)*2/2;
    then
A50: 0+b>t`1 by XREAL_1:19;
    -1<A*(t`1)+B by A38,A47,A45,EUCLID:52;
    then (-1)*(b-a)< (2*((t`1)-b)-(a-b))/(b-a)*(b-a) by A26,A44,XREAL_1:68;
    then (-1)*(b-a)< 2*((t`1)-b)-(a-b) by A26,XCMPLX_1:87;
    then (-1)*(b-a)+(a-b)< 2*((t`1)-b)-(a-b)+(a-b) by XREAL_1:8;
    then 2*(a-b)/2< 2*((t`1)-b)/2 by XREAL_1:74;
    then a < (t`1) by XREAL_1:9;
    hence contradiction by A17,A50,A48,A49;
  end;
A51: dom f=the carrier of I[01] by FUNCT_2:def 1;
  then
A52: f2.I=ff.(f.I) by FUNCT_1:13
    .=|[A*b+B,C*((f.I)`2)+D]| by A4,Def2;
  then
A53: (f2.I)`1=A*b+B by EUCLID:52
    .= b*2/(b-a)+(1-2*b/(b-a))
    .= 1;
A54: f2.O=ff.(f.O) by A51,FUNCT_1:13
    .=|[A*a+B,C*((f.O)`2)+D]| by A3,Def2;
  then
A55: (f2.O)`1=A*a+B by EUCLID:52
    .= a*2/(b-a)+(1-2*b/(b-a))
    .= a*2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A26,XCMPLX_1:60
    .= a*2/(b-a)+((b-a)-2*b)/(b-a)
    .= (a*2+((b-a)-2*b))/(b-a)
    .= (-(b-a))/(b-a)
    .= -((b-a)/(b-a))
    .= -1 by A26,XCMPLX_1:60;
A56: now
    assume rng g2 meets K0;
    then consider x being object such that
A57: x in rng g2 and
A58: x in K0 by XBOOLE_0:3;
    reconsider q=x as Point of TOP-REAL 2 by A57;
    consider p such that
A59: p=q and
A60: -1<p`1 and
A61: p`1<1 and
A62: -1<p`2 and
A63: p`2<1 by A58;
    consider z being object such that
A64: z in dom g2 and
A65: x=g2.z by A57,FUNCT_1:def 3;
    reconsider u=z as Point of I[01] by A64;
    reconsider t=g.u as Point of TOP-REAL 2;
A66: A*(t`1)+B= (t`1)*2/(b-a)+(1-2*b/(b-a))
      .= (t`1)*2/(b-a)+((b-a)/(b-a)-2*b/(b-a)) by A26,XCMPLX_1:60
      .= (t`1)*2/(b-a)+((b-a)-2*b)/(b-a)
      .= ((t`1)*2+((b-a)-2*b))/(b-a)
      .= (2*((t`1)-b)-(a-b))/(b-a);
A67: ff.t=p by A59,A64,A65,FUNCT_1:12;
A68: C*(t`2)+D= (t`2)*2/(d-c)+(1-2*d/(d-c))
      .= (t`2)*2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A19,XCMPLX_1:60
      .= (t`2)*2/(d-c)+((d-c)-2*d)/(d-c)
      .= ((t`2)*2+((d-c)-2*d))/(d-c)
      .= (2*((t`2)-d)-(c-d))/(d-c);
A69: ff.t=|[A*(t`1)+B,C*(t`2)+D]| by Def2;
    then -1<C*(t`2)+D by A62,A67,EUCLID:52;
    then (-1)*(d-c)< (2*((t`2)-d)-(c-d))/(d-c)*(d-c) by A19,A68,XREAL_1:68;
    then (-1)*(d-c)< 2*((t`2)-d)-(c-d) by A19,XCMPLX_1:87;
    then (-1)*(d-c)+(c-d)< 2*((t`2)-d)-(c-d)+(c-d) by XREAL_1:8;
    then 2*(c-d)/2< 2*((t`2)-d)/2 by XREAL_1:74;
    then
A70: c < (t`2) by XREAL_1:9;
    C*(t`2)+D<1 by A63,A69,A67,EUCLID:52;
    then (1)*(d-c)> (2*((t`2)-d)-(c-d))/(d-c)*(d-c) by A19,A68,XREAL_1:68;
    then (1)*(d-c)> 2*((t`2)-d)-(c-d) by A19,XCMPLX_1:87;
    then (1)*(d-c)+(c-d)> 2*((t`2)-d)-(c-d)+(c-d) by XREAL_1:8;
    then 0/2>((t`2)-d)*2/2;
    then
A71: 0+d>t`2 by XREAL_1:19;
    A*(t`1)+B<1 by A61,A69,A67,EUCLID:52;
    then (1)*(b-a)> (2*((t`1)-b)-(a-b))/(b-a)*(b-a) by A26,A66,XREAL_1:68;
    then (1)*(b-a)> 2*((t`1)-b)-(a-b) by A26,XCMPLX_1:87;
    then (1)*(b-a)+(a-b)> 2*((t`1)-b)-(a-b)+(a-b) by XREAL_1:8;
    then 0/2>((t`1)-b)*2/2;
    then
A72: 0+b>t`1 by XREAL_1:19;
    -1<A*(t`1)+B by A60,A69,A67,EUCLID:52;
    then (-1)*(b-a)< (2*((t`1)-b)-(a-b))/(b-a)*(b-a) by A26,A66,XREAL_1:68;
    then (-1)*(b-a)< 2*((t`1)-b)-(a-b) by A26,XCMPLX_1:87;
    then (-1)*(b-a)+(a-b)< 2*((t`1)-b)-(a-b)+(a-b) by XREAL_1:8;
    then 2*(a-b)/2< 2*((t`1)-b)/2 by XREAL_1:74;
    then a < (t`1) by XREAL_1:9;
    hence contradiction by A18,A72,A70,A71;
  end;
A73: -1<=(f2.O)`2 & (f2.O)`2<=1 & -1<=(f2.I)`2 & (f2.I)`2<=1
  proof
    reconsider s1=(f.I)`2 as Real;
    reconsider s0=(f.O)`2 as Real;
A74: (c-d)/(d-c) = (-(d-c))/(d-c) .= -((d-c)/(d-c))
      .= -1 by A19,XCMPLX_1:60;
A75: (f2.I)`2=C*s1+D by A52,EUCLID:52
      .= s1*2/(d-c)+(1-2*d/(d-c))
      .= s1*2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A19,XCMPLX_1:60
      .= s1*2/(d-c)+((d-c)-2*d)/(d-c)
      .= (s1*2+((d-c)-2*d))/(d-c)
      .= ((s1-d)+(s1-d)-(c-d))/(d-c);
    d-d>=s0-d by A6,XREAL_1:9;
    then 0+(d-d)-(c-d)>=(s0-d)+(s0-d)-(c-d) by XREAL_1:9;
    then
A76: (d-c)/(d-c)>=((s0-d)+(s0-d)-(c-d))/(d-c) by A19,XREAL_1:72;
    d-d>=s1-d by A8,XREAL_1:9;
    then
A77: 0+(d-d)-(c-d)>=(s1-d)+(s1-d)-(c-d) by XREAL_1:9;
    c-d<=s1-d by A7,XREAL_1:9;
    then c-d+(c-d)<=(s1-d)+(s1-d) by XREAL_1:7;
    then
A78: c-d+(c-d)-(c-d)<=(s1-d)+(s1-d)-(c-d) by XREAL_1:9;
    c-d<=s0-d by A5,XREAL_1:9;
    then c-d+(c-d)<=(s0-d)+(s0-d) by XREAL_1:7;
    then
A79: c-d+(c-d)-(c-d)<=(s0-d)+(s0-d)-(c-d) by XREAL_1:9;
    (f2.O)`2=C*s0+D by A54,EUCLID:52
      .= s0 *2/(d-c)+(1-2*d/(d-c))
      .= s0 *2/(d-c)+((d-c)/(d-c)-2*d/(d-c)) by A19,XCMPLX_1:60
      .= s0 *2/(d-c)+((d-c)-2*d)/(d-c)
      .= (s0 *2+((d-c)-2*d))/(d-c)
      .= ((s0-d)+(s0-d)-(c-d))/(d-c);
    hence thesis by A19,A79,A74,A76,A75,A78,A77,XREAL_1:72;
  end;
  set y = the Element of rng f2 /\ rng g2;
  A>0 by A26,XREAL_1:139;
  then
A80: ff is one-to-one by A20,Th44;
  then f2 is one-to-one & g2 is one-to-one by A2,FUNCT_1:24;
  then rng f2 meets rng g2 by A1,A2,A55,A53,A25,A23,A73,A27,A34,A56,Th42;
  then
A81: rng f2 /\ rng g2 <> {};
  then y in rng f2 by XBOOLE_0:def 4;
  then consider x being object such that
A82: x in dom f2 and
A83: y=f2.x by FUNCT_1:def 3;
  dom f2 c= dom f by RELAT_1:25;
  then
A84: f.x in rng f by A82,FUNCT_1:3;
  y in rng g2 by A81,XBOOLE_0:def 4;
  then consider x2 being object such that
A85: x2 in dom g2 and
A86: y=g2.x2 by FUNCT_1:def 3;
A87: y=ff.(g.x2) by A85,A86,FUNCT_1:12;
  dom g2 c= dom g by RELAT_1:25;
  then
A88: g.x2 in rng g by A85,FUNCT_1:3;
  dom ff=the carrier of TOP-REAL 2 & y=ff.(f.x) by A82,A83,FUNCT_1:12
,FUNCT_2:def 1;
  then f.x=g.x2 by A80,A87,A84,A88,FUNCT_1:def 4;
  then rng f /\ rng g <> {} by A84,A88,XBOOLE_0:def 4;
  hence thesis;
end;
