A1: dom s = the carrier of SCMPDS by PARTFUN1:def 2;

now

let x be set ; :: thesis: ( x in dom (s +* (li,k)) implies (s +* (li,k)) . b₁ in the Object-Kind of SCMPDS . b₁ )
assume x in dom (s +* (li,k)) ; :: thesis: (s +* (li,k)) . b₁ in the Object-Kind of SCMPDS . b₁
then A2: x in dom s by A1, PARTFUN1:def 2;

per cases ( x = li or x <> li ) ;

suppose A3: x = li ; :: thesis: (s +* (li,k)) . b₁ in the Object-Kind of SCMPDS . b₁

then A4: the Object-Kind of SCMPDS . x = ObjectKind li
.= INT by SCMPDS_2:5 ;
(s +* (li,k)) . x = k by A1, A3, FUNCT_7:31;
hence (s +* (li,k)) . x in the Object-Kind of SCMPDS . x by A4, INT_1:def 2; :: thesis: verum

end;

suppose x <> li ; :: thesis: (s +* (li,k)) . b₁ in the Object-Kind of SCMPDS . b₁

then (s +* (li,k)) . x = s . x by FUNCT_7:32;
hence (s +* (li,k)) . x in the Object-Kind of SCMPDS . x by A2, FUNCT_1:def 14; :: thesis: verum

end;

end;

end;

hence s +* (li,k) is PartState of SCMPDS by FUNCT_1:def 14; :: thesis: verum