reserve p,q for Point of TOP-REAL 2;

theorem Th13:
  for f,g being Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN
  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 & C0={p: |.p.|<=1}& KXP={q1
where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} & KXN={q2
where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} & KYP={q3
where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} & KYN={q4
where q4 is Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} & f.O in
  KXN & f.I in KXP & g.O in KYP & g.I in KYN & rng f c= C0 & rng g c= C0 holds
  rng f meets rng g
proof
  let f,g be Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN be Subset of
  TOP-REAL 2, O,I be Point of I[01];
  assume
A1: O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one
& C0={p: |.p.|<=1}& KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=
q1`1 & q1`2>=-q1`1} & KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2
>=q2`1 & q2`2<=-q2`1} & KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3
  `2>=q3`1 & q3`2>=-q3`1} & KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 &
q4`2<=q4`1 & q4`2<=-q4`1} & f.O in KXN & f.I in KXP & g.O in KYP & g.I in KYN &
  rng f c= C0 & rng g c= C0;
  then
  ex g2 being Function of I[01],TOP-REAL 2 st g2.0=g.1 & g2 .1=g.0 & rng g2
  =rng g & g2 is continuous one-to-one by Th12;
  hence thesis by A1,JGRAPH_3:44;
end;
