 reserve h,h1 for 0-convergent non-zero Real_Sequence,
         c,c1 for constant Real_Sequence;

theorem Th18: ::: generalized XREAL_1:5
  for a,b be R_eal st a < b ex c be Real st a < c & c < b
proof
    let a,b be R_eal;
    assume A1: a < b;
    per cases;
    suppose A2: a in REAL; then
A3:  b in REAL or b = +infty by A1,XXREAL_0:10;

     now assume A4: b = +infty;
      consider c be Real such that
A5:    a < c by A2,XREAL_1:1;
      c in REAL by XREAL_0:def 1;
      hence ex c be Real st a < c & c < b by A5,A4,XXREAL_0:9;
     end;
     hence thesis by A3,A1,A2,XREAL_1:5;
    end;
    suppose A6: b in REAL; then
A7:  a in REAL or a = -infty by A1,XXREAL_0:13;

     now assume A8: a = -infty;
      consider c be Real such that
A9:    c < b by A6,XREAL_1:2;
      c in REAL by XREAL_0:def 1;
      hence ex c be Real st a < c & c < b by A9,A8,XXREAL_0:12;
     end;
     hence thesis by A7,A1,A6,XREAL_1:5;
    end;
    suppose not a in REAL & not b in REAL; then
     a = +infty or a = -infty &
     b = +infty or b = -infty by XXREAL_0:14; then
     a < 0 & 0 < b by A1,XXREAL_0:4,6;
     hence ex c be Real st a < c & c < b;
    end;
end;
