
theorem Th24:
for a,b be Real, I be Subset of R^1 st I = [.a,b.] holds I is compact
proof
    let a,b be Real, I be Subset of R^1;
    assume A1: I = [.a,b.];
    per cases;
    suppose A2: a <= b; then
     Closed-Interval-TSpace(a,b) is compact by HEINE:4; then
A3:  [#]Closed-Interval-TSpace(a,b) is compact by COMPTS_1:1;
     [#]Closed-Interval-TSpace(a,b)
      = the carrier of Closed-Interval-TSpace(a,b) by STRUCT_0:def 3; then
     I = [#]Closed-Interval-TSpace(a,b) by A1,A2,TOPMETR:18;
     hence I is compact by A3,COMPTS_1:19;
    end;
    suppose a > b; then
     [.a,b.] = {} by XXREAL_1:29;
     hence I is compact by A1;
    end;
end;
