reserve a,b,c,d for Real;
reserve r,s for Real;
reserve p,q,r,s,t for ExtReal;

theorem Th227:
  r < t implies ex s st r < s & s < t
proof
  assume
A1: r < t;
  then
A2: r <> +infty by XXREAL_0:3;
A3: t <> -infty by A1,XXREAL_0:5;
  per cases by A2,A3,XXREAL_0:14;
  suppose that
A4: r = -infty and
A5: t = +infty;
    take 0;
    thus thesis by A4,A5;
  end;
  suppose that
A6: r = -infty and
A7: t in REAL;
    reconsider t as Real by A7;
    consider s being Real such that
A8: s < t by Th2;
    take s;
    s in REAL by XREAL_0:def 1;
    hence r < s by A6,XXREAL_0:12;
    thus thesis by A8;
  end;
  suppose that
A9: r in REAL and
A10: t = +infty;
    reconsider r9 = r as Real by A9;
    consider s being Real such that
A11: r9 < s by Th1;
    take s;
    thus r < s by A11;
    s in REAL by XREAL_0:def 1;
    hence thesis by A10,XXREAL_0:9;
  end;
  suppose that
A12: r in REAL and
A13: t in REAL;
    reconsider r,t as Real by A12,A13;
    consider s being Real such that
A14: r < s & s < t by A1,Th5;
    take s;
    thus thesis by A14;
  end;
end;
