let A, B be ext-real-membered set ; :: thesis: ( A is connected & B is connected & A meets B implies A \/ B is connected )
assume that
Z1: A is connected and
Z2: B is connected ; :: thesis: ( not A meets B or A \/ B is connected )
given z being ext-real number such that G1: z in A and
G2: z in B ; :: according to MEMBERED:def 20 :: thesis: A \/ B is connected
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 <= z or z <= r ) ;

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

then r in B by Z2, S2, G2, Z6, Th84;
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 ( z <= r or r <= z ) ;

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

then r in B by Z2, S1, G2, Z5, 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 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 Th89; :: thesis: verum