let X, Y be set ; :: thesis: UsedI*Loc (X \/ Y) = (UsedI*Loc X) \/ (UsedI*Loc Y)
thus UsedI*Loc (X \/ Y) c= (UsedI*Loc X) \/ (UsedI*Loc Y) :: according to XBOOLE_0:def 10 :: thesis: (UsedI*Loc X) \/ (UsedI*Loc Y) c= UsedI*Loc (X \/ Y)

proof

let e be object ; :: according to TARSKI:def 3 :: thesis: ( not e in UsedI*Loc (X \/ Y) or e in (UsedI*Loc X) \/ (UsedI*Loc Y) )
assume e in UsedI*Loc (X \/ Y) ; :: thesis: e in (UsedI*Loc X) \/ (UsedI*Loc Y)
then consider B being set such that
A1: e in B and
A2: B in { (UsedInt*Loc i) where i is Instruction of SCM+FSA : i in X \/ Y } by TARSKI:def 4;
consider i being Instruction of SCM+FSA such that
A3: B = UsedInt*Loc i and
A4: i in X \/ Y by A2;

now :: thesis: ( ( i in X & e in UsedI*Loc X ) or ( i in Y & e in UsedI*Loc Y ) )

per cases ( i in X or i in Y ) by A4, XBOOLE_0:def 3;

case i in X ; :: thesis: e in UsedI*Loc X

then B in { (UsedInt*Loc j) where j is Instruction of SCM+FSA : j in X } by A3;
hence e in UsedI*Loc X by A1, TARSKI:def 4; :: thesis: verum

end;

case i in Y ; :: thesis: e in UsedI*Loc Y

then B in { (UsedInt*Loc j) where j is Instruction of SCM+FSA : j in Y } by A3;
hence e in UsedI*Loc Y by A1, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence e in (UsedI*Loc X) \/ (UsedI*Loc Y) by XBOOLE_0:def 3; :: thesis: verum

end;

( X c= X \/ Y & Y c= X \/ Y ) by XBOOLE_1:7;
then ( UsedI*Loc X c= UsedI*Loc (X \/ Y) & UsedI*Loc Y c= UsedI*Loc (X \/ Y) ) by Lm5;
hence (UsedI*Loc X) \/ (UsedI*Loc Y) c= UsedI*Loc (X \/ Y) by XBOOLE_1:8; :: thesis: verum