reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th75:
  for A being connected Subset of R^1, a, b, c being Real
  st a <= b & b <= c & a in A & c in A holds b in A
proof
  let A be connected Subset of R^1, a, b, c be Real;
  assume that
A1: a <= b and
A2: b <= c and
A3: a in A and
A4: c in A;
  per cases by A1,A2,A3,A4,XXREAL_0:1;
  suppose
    a = b or b = c;
    hence thesis by A3,A4;
  end;
  suppose
A5: a < b & b < c & a in A & c in A;
    reconsider B = ]. -infty,b .[, C = ]. b,+infty .[ as Subset of R^1 by
TOPMETR:17;
    assume not b in A;
    then A c= REAL \ {b} by TOPMETR:17,ZFMISC_1:34;
    then
A6: A c= ]. -infty,b .[ \/ ]. b,+infty .[ by XXREAL_1:389;
    now
      per cases by A6,Th52,CONNSP_1:16;
      suppose
        A c= B;
        hence contradiction by A5,XXREAL_1:233;
      end;
      suppose
        A c= C;
        hence contradiction by A5,XXREAL_1:235;
      end;
    end;
    hence thesis;
  end;
end;
