let A, B be ext-real-membered set ; :: thesis: ( A is right_end & A is connected & B is connected & sup A = inf B implies A \/ B is connected )
assume that
Z1: ( A is right_end & A is connected ) and
Z2: B is connected and
Z3: sup A = inf B ; :: thesis: A \/ B is connected
set z = inf B;
G1: inf B in A by Z1, Def6, Z3;
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
Z3: x in A \/ B and
Z4: y in A \/ B and
Z5: x < r and
Z6: 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 Z3, Z4, XBOOLE_0:def 3;

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

then r in A by Z1, Z5, Z6, Th84;
hence r in A \/ B by XBOOLE_0:def 3; :: thesis: verum

end;

suppose that S1: x in A and
S2: 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 Z1, S1, G1, Z5, Th84;
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 Z2, S2, Z6, Th86;
hence r in A \/ B by XBOOLE_0:def 3; :: thesis: verum

end;

end;

end;

suppose that S1: x in B and
S2: y in A ; :: thesis: r in A \/ B

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

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

then r in A by Z1, S2, G1, Z6, Th84;
hence r in A \/ B by XBOOLE_0:def 3; :: thesis: verum

end;

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

inf B <= x by S1, Th3;
hence r in A \/ B by S, Z5, XXREAL_0:2; :: thesis: verum

end;

end;

end;

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

then r in B by Z2, Z5, Z6, Th84;
hence r in A \/ B by XBOOLE_0:def 3; :: thesis: verum

end;

end;

end;

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