
theorem Th9:
  for P being Subset of TOP-REAL 2, p1,p2,q1 being Point of
  TOP-REAL 2 st q1 in P holds LE q1,q1,P,p1,p2
proof
  let P be Subset of TOP-REAL 2, p1,p2,q1 be Point of TOP-REAL 2;
  assume
A1: q1 in P;
  then reconsider P as non empty Subset of TOP-REAL 2;
  now
    let g be Function of I[01], (TOP-REAL 2)|P, s1,s2 be Real;
    assume that
A2: g is being_homeomorphism and
    g.0=p1 and
    g.1=p2 and
A3: g.s1=q1 and
A4: 0<=s1 & s1<=1 and
A5: g.s2=q1 and
A6: 0<=s2 & s2<=1;
A7: g is one-to-one by A2,TOPS_2:def 5;
    s1 in the carrier of I[01] & s2 in the carrier of I[01] by A4,A6,
BORSUK_1:43;
    hence s1 <= s2 by A3,A5,A7,FUNCT_2:19;
  end;
  hence thesis by A1;
end;
