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 & Y c= X holds
f | Y is constant
let C, D be non empty set ; :: thesis: for f being PartFunc of C,D st f | X is constant & Y c= X holds
f | Y is constant
let f be PartFunc of C,D; :: thesis: ( f | X is constant & Y c= X implies f | Y is constant )
assume that
A1: f | X is constant and
A2: Y c= X ; :: thesis: f | Y is constant
consider d being Element of D such that
A3: for c being Element of C st c in X /\ (dom f) holds
f /. c = d by A1, Th35;

now :: thesis: for c being Element of C st c in Y /\ (dom f) holds
f /. c = d

let c be Element of C; :: thesis: ( c in Y /\ (dom f) implies f /. c = d )
assume c in Y /\ (dom f) ; :: thesis: f /. c = d
then ( c in Y & c in dom f ) by XBOOLE_0:def 4;
then c in X /\ (dom f) by A2, XBOOLE_0:def 4;
hence f /. c = d by A3; :: thesis: verum

end;

hence f | Y is constant by Th35; :: thesis: verum