reserve x, y for set;
reserve i, j, n for Nat;
reserve p1, p2 for Point of TOP-REAL n;
reserve a, b, c, d for Real;

theorem Th3:
  for g being Path of p1,p2 st rng g c= LSeg(p1,p2) holds rng g = LSeg(p1,p2)
proof
  let g be Path of p1,p2;
  assume
A1: rng g c= LSeg(p1,p2);
A2: p2=g.1 by BORSUK_2:def 4;
A3: p1=g.0 by BORSUK_2:def 4;
  now
A4: g.:([#]I[01]) c= rng g
    proof
      let y be object;
      assume y in g.:([#]I[01]);
      then ex x being object st x in dom g & x in [#]I[01] & y=g.x by
FUNCT_1:def 6;
      hence thesis by FUNCT_1:def 3;
    end;
    1 in the carrier of I[01] by BORSUK_1:43;
    then 1 in dom g by FUNCT_2:def 1;
    then
A5: p2 in g.:([#]I[01]) by A2,FUNCT_1:def 6;
    0 in the carrier of I[01] by BORSUK_1:43;
    then 0 in dom g by FUNCT_2:def 1;
    then
A6: p1 in g.:([#]I[01]) by A3,FUNCT_1:def 6;
    [#]I[01] is connected by CONNSP_1:27;
    then
A7: g.:([#]I[01]) is connected by TOPS_2:61;
    assume not LSeg(p1,p2) c= rng g;
    hence contradiction by A1,A7,A6,A5,A4,Th2,XBOOLE_1:1;
  end;
  hence thesis by A1;
end;
