A1: dom (s +* li,k) = dom s by FUNCT_7:32;
A2: dom s = dom the Object-Kind of SCM+FSA by CARD_3:18;

now

let x be set ; :: thesis: ( x in dom the Object-Kind of SCM+FSA implies (s +* li,k) . b₁ in the Object-Kind of SCM+FSA . b₁ )
assume A3: x in dom the Object-Kind of SCM+FSA ; :: thesis: (s +* li,k) . b₁ in the Object-Kind of SCM+FSA . b₁

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

suppose A4: x = li ; :: thesis: (s +* li,k) . b₁ in the Object-Kind of SCM+FSA . b₁

then A5: (s +* li,k) . x = k by A2, A3, FUNCT_7:33;
the Object-Kind of SCM+FSA . x = ObjectKind li by A4
.= INT by SCMFSA_2:26 ;
hence (s +* li,k) . x in the Object-Kind of SCM+FSA . x by A5, INT_1:def 2; :: thesis: verum

end;

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

then (s +* li,k) . x = s . x by FUNCT_7:34;
hence (s +* li,k) . x in the Object-Kind of SCM+FSA . x by A3, CARD_3:18; :: thesis: verum

end;

end;

end;

hence s +* li,k is State of SCM+FSA by A1, A2, CARD_3:18; :: thesis: verum