let A, B be ext-real-membered set ; :: thesis: ( A is interval & B is left_end & B is interval & sup A = inf B implies A \/ B is interval )
assume that
A1: A is interval and
A2: ( B is left_end & B is interval ) and
A3: sup A = inf B ; :: thesis: A \/ B is interval
set z = inf B;
A4: inf B in B by A2, Def5;
for x, y, r being ext-real number st x in A \/ B & y in A \/ B & x < r & r < y holds
r in A \/ B

proof

let x, y, r be ext-real number ; :: thesis: ( x in A \/ B & y in A \/ B & x < r & r < y implies r in A \/ B )
assume that
A5: x in A \/ B and
A6: y in A \/ B and
A7: x < r and
A8: r < y ; :: thesis: r in A \/ B

per cases ( ( x in A & y in A ) or ( x in A & y in B ) or ( x in B & y in A ) or ( x in B & y in B ) ) by A5, A6, XBOOLE_0:def 3;

suppose ( x in A & y in A ) ; :: thesis: r in A \/ B

then r in A by A1, A7, A8, Th84;
hence r in A \/ B by XBOOLE_0:def 3; :: thesis: verum

end;

suppose that A9: x in A and
A10: y in B ; :: thesis: r in A \/ B

per cases ( r < inf B or inf B <= r ) ;

suppose r < inf B ; :: thesis: r in A \/ B

then r in A by A1, A3, A7, A9, Th85;
hence r in A \/ B by XBOOLE_0:def 3; :: thesis: verum

end;

suppose inf B <= r ; :: thesis: r in A \/ B

then r in B by A2, A4, A8, A10, Th84;
hence r in A \/ B by XBOOLE_0:def 3; :: thesis: verum

end;

end;

end;

suppose that A11: x in B and
A12: y in A ; :: thesis: r in A \/ B

per cases ( inf B < r or r <= inf B ) ;

suppose A13: inf B < r ; :: thesis: r in A \/ B

y <= inf B by A3, A12, Th4;
hence r in A \/ B by A8, A13, XXREAL_0:2; :: thesis: verum

end;

suppose r <= inf B ; :: thesis: r in A \/ B

then r in B by A2, A4, A7, A11, Th84;
hence r in A \/ B by XBOOLE_0:def 3; :: thesis: verum

end;

end;

end;

suppose ( x in B & y in B ) ; :: thesis: r in A \/ B

then r in B by A2, A7, A8, Th84;
hence r in A \/ B by XBOOLE_0:def 3; :: thesis: verum

end;

end;

end;

hence A \/ B is interval by Th88; :: thesis: verum