
theorem Th10:
  for P being Subset of TOP-REAL 2, p1,p2,q1 being Point of
TOP-REAL 2 st P is_an_arc_of p1,p2 & q1 in P holds LE p1,q1,P,p1,p2 & LE q1,p2,
  P,p1,p2
proof
  let P be Subset of TOP-REAL 2, p1,p2,q1 be Point of TOP-REAL 2;
  assume that
A1: P is_an_arc_of p1,p2 and
A2: q1 in P;
  reconsider P as non empty Subset of TOP-REAL 2 by A2;
A3: now
A4: 0 in the carrier of I[01] by BORSUK_1:43;
    let g be Function of I[01], (TOP-REAL 2)|P,s1,s2 be Real;
    assume that
A5: g is being_homeomorphism and
A6: g.0=p1 and
    g.1=p2 and
A7: g.s1=p1 and
A8: 0<=s1 & s1<=1 and
    g.s2=q1 and
A9: 0<=s2 and
    s2<=1;
    s1 in the carrier of I[01] & g is one-to-one by A5,A8,BORSUK_1:43
,TOPS_2:def 5;
    hence s1 <= s2 by A6,A7,A9,A4,FUNCT_2:19;
  end;
A10: now
A11: 1 in the carrier of I[01] by BORSUK_1:43;
    let g be Function of I[01], (TOP-REAL 2)|P,s1,s2 be Real;
    assume that
A12: g is being_homeomorphism and
    g.0=p1 and
A13: g.1=p2 and
    g.s1=q1 and
    0<=s1 and
A14: s1<=1 & g.s2=p2 and
A15: 0<=s2 & s2<=1;
    s2 in the carrier of I[01] & g is one-to-one by A12,A15,BORSUK_1:43
,TOPS_2:def 5;
    hence s1 <= s2 by A13,A14,A11,FUNCT_2:19;
  end;
  p1 in P & p2 in P by A1,TOPREAL1:1;
  hence thesis by A2,A3,A10;
end;
