let X, Y be set ; :: thesis: for C, D being non empty set
for f being PartFunc of C,D st f | X is constant & f | Y is constant & X /\ Y meets dom f holds
f | (X \/ Y) is constant
let C, D be non empty set ; :: thesis: for f being PartFunc of C,D st f | X is constant & f | Y is constant & X /\ Y meets dom f holds
f | (X \/ Y) is constant
let f be PartFunc of C,D; :: thesis: ( f | X is constant & f | Y is constant & X /\ Y meets dom f implies f | (X \/ Y) is constant )
assume that
A1: f | X is constant and
A2: f | Y is constant and
A3: (X /\ Y) /\ (dom f) <> {} ; :: according to XBOOLE_0:def 7 :: thesis: f | (X \/ Y) is constant
consider d1 being Element of D such that
A4: for c being Element of C st c in X /\ (dom f) holds
f /. c = d1 by A1, Th35;
set x = the Element of (X /\ Y) /\ (dom f);
A5: the Element of (X /\ Y) /\ (dom f) in X /\ Y by A3, XBOOLE_0:def 4;
A6: the Element of (X /\ Y) /\ (dom f) in dom f by A3, XBOOLE_0:def 4;
then reconsider x = the Element of (X /\ Y) /\ (dom f) as Element of C ;
x in Y by A5, XBOOLE_0:def 4;
then A7: x in Y /\ (dom f) by A6, XBOOLE_0:def 4;
consider d2 being Element of D such that
A8: for c being Element of C st c in Y /\ (dom f) holds
f /. c = d2 by A2, Th35;
x in X by A5, XBOOLE_0:def 4;
then x in X /\ (dom f) by A6, XBOOLE_0:def 4;
then f /. x = d1 by A4;
then A9: d1 = d2 by A8, A7;
take d1 ; :: according to PARTFUN2:def 1 :: thesis: for c being Element of C st c in dom (f | (X \/ Y)) holds
(f | (X \/ Y)) . c = d1
let c be Element of C; :: thesis: ( c in dom (f | (X \/ Y)) implies (f | (X \/ Y)) . c = d1 )
assume A10: c in dom (f | (X \/ Y)) ; :: thesis: (f | (X \/ Y)) . c = d1
then A11: c in (X \/ Y) /\ (dom f) by RELAT_1:61;
then A12: c in dom f by XBOOLE_0:def 4;
A13: c in X \/ Y by A11, XBOOLE_0:def 4;

now :: thesis: f /. c = d1

per cases ( c in X or c in Y ) by A13, XBOOLE_0:def 3;

suppose c in X ; :: thesis: f /. c = d1

then c in X /\ (dom f) by A12, XBOOLE_0:def 4;
hence f /. c = d1 by A4; :: thesis: verum

end;

suppose c in Y ; :: thesis: f /. c = d1

then c in Y /\ (dom f) by A12, XBOOLE_0:def 4;
hence f /. c = d1 by A8, A9; :: thesis: verum

end;

end;

end;

then (f | (X \/ Y)) /. c = d1 by A11, Th16;
hence (f | (X \/ Y)) . c = d1 by A10, PARTFUN1:def 6; :: thesis: verum