
theorem Th11:
  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 & a <> b & c <> d & (f.O)`1=a & c <=(f.O)`2 & (f.O)`2 <=d
& (f.I)`1=b & c <=(f.I)`2 & (f.I)`2 <=d & (g.O)`2=c & a <=(g.O)`1 & (g.O)`1 <=b
& (g.I)`2=d & a <=(g.I)`1 & (g.I)`1 <=b & (for r being Point of I[01] holds (a
>=(f.r)`1 or (f.r)`1>=b or c >=(f.r)`2 or (f.r)`2>=d) & (a >=(g.r)`1 or (g.r)`1
  >=b or c >=(g.r)`2 or (g.r)`2>=d)) holds rng f meets rng g
proof
  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 & f is continuous one-to-one & g is continuous one-to-one and
A2: a <> b and
A3: c <> d and
A4: (f.O)`1=a and
A5: c <=(f.O)`2 & (f.O)`2 <=d and
A6: (f.I)`1=b & c <=(f.I)`2 & (f.I)`2 <=d & (g.O)`2=c and
A7: a <=(g.O)`1 & (g.O)`1 <=b and
A8: (g.I)`2=d & a <=(g.I)`1 &( (g.I)`1 <=b & for r being Point of I[01]
  holds (a >=(f.r)`1 or (f.r)`1>=b or c >=(f.r)`2 or (f.r)`2>=d) & (a >=(g.r)`1
  or (g.r)`1>=b or c >=(g.r)`2 or (g.r) `2 >=d) );
  c <= d by A5,XXREAL_0:2;
  then
A9: c < d by A3,XXREAL_0:1;
  a <= b by A7,XXREAL_0:2;
  then a < b by A2,XXREAL_0:1;
  hence thesis by A1,A4,A5,A6,A7,A8,A9,JGRAPH_2:45;
end;
