reserve a, b, r, s for Real;

theorem Th39:
  for X being Subset of R^1 st X = ].a,b.] holds Int X = ].a,b.[
proof
  let X be Subset of R^1 such that
A1: X = ].a,b.];
A2: Int X c= X by TOPS_1:16;
  thus Int X c= ].a,b.[
  proof
    let x be object;
    assume
A3: x in Int X;
    then reconsider x as Point of R^1;
A4: now
      now
        assume a >= b;
        then X = {}R^1 by A1,XXREAL_1:26;
        hence contradiction by A3;
      end;
      then Fr X = {a,b} by A1,Th35;
      then
A5:   Int X misses Fr X & b in Fr X by TARSKI:def 2,TOPS_1:39;
      assume x = b;
      hence contradiction by A3,A5,XBOOLE_0:3;
    end;
    x <= b by A1,A2,A3,XXREAL_1:2;
    then
A6: x < b by A4,XXREAL_0:1;
    a < x by A1,A2,A3,XXREAL_1:2;
    hence thesis by A6,XXREAL_1:4;
  end;
  reconsider Y = ].a,b.[ as open Subset of R^1 by BORSUK_5:39,TOPMETR:17;
  Y c= Int X by A1,TOPS_1:24,XXREAL_1:41;
  hence thesis;
end;
