A1: now

let x be set ; :: thesis: ( x in dom (SCM-OK R) implies (s +* (t .--> u)) . x in (SCM-OK R) . x )
assume A2: x in dom (SCM-OK R) ; :: thesis: (s +* (t .--> u)) . x in (SCM-OK R) . x

now

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

suppose A3: x = t ; :: thesis: (s +* (t .--> u)) . x in (SCM-OK R) . x

{t} = dom (t .--> u) by FUNCOP_1:13;
then t in dom (t .--> u) by TARSKI:def 1;
then (s +* (t .--> u)) . t = (t .--> u) . t by FUNCT_4:13
.= u by FUNCOP_1:72 ;
then (s +* (t .--> u)) . t in the carrier of R ;
hence (s +* (t .--> u)) . x in (SCM-OK R) . x by A3, Th3; :: thesis: verum

end;

suppose A4: x <> t ; :: thesis: (s +* (t .--> u)) . x in (SCM-OK R) . x

{t} = dom (t .--> u) by FUNCOP_1:13;
then not x in dom (t .--> u) by A4, TARSKI:def 1;
then (s +* (t .--> u)) . x = s . x by FUNCT_4:11;
hence (s +* (t .--> u)) . x in (SCM-OK R) . x by A2, CARD_3:9; :: thesis: verum

end;

end;

end;

hence (s +* (t .--> u)) . x in (SCM-OK R) . x ; :: thesis: verum

end;

A5: dom (SCM-OK R) = SCM-Memory by FUNCT_2:def 1;
then dom s = SCM-Memory by CARD_3:9;
then dom (s +* (t .--> u)) = SCM-Memory \/ (dom (t .--> u)) by FUNCT_4:def 1
.= SCM-Memory \/ {t} by FUNCOP_1:13
.= dom (SCM-OK R) by A5, ZFMISC_1:40 ;
hence s +* (t .--> u) is SCM-State of R by A1, CARD_3:9; :: thesis: verum