
theorem Th2:
for I,J,K be closed_interval Subset of REAL,
 E be Subset of [:[:RNS_Real,RNS_Real:],RNS_Real:] st E = [:[:I,J:],K:]
 holds E is compact
proof
    let I,J,K be closed_interval Subset of REAL,
    E be Subset of [:[:RNS_Real,RNS_Real:],RNS_Real:];
    assume
A1: E = [:[:I,J:],K:];

    set S = [:[:RNS_Real,RNS_Real:],RNS_Real:];

    reconsider AB = [:I,J:] as Subset of [:RNS_Real,RNS_Real:];
    reconsider C = K as Subset of RNS_Real;
    AB is compact & C is compact by MESFUN16:7,9;
    hence E is compact by A1,MESFUN16:8;
end;
