defpred S₁[ set , set ] means ( ( $1 = NAT & $2 = NAT ) or ( $1 in SCM-Data-Loc & $2 = the carrier of S ) );

A1: now

let k be Element of SCM-Memory ; :: thesis: ex b being Element of { the carrier of S} \/ {NAT} st S₁[k,b]
A2: { the carrier of S} \/ {NAT} = { the carrier of S,NAT} by ENUMSET1:1;
then A3: NAT in { the carrier of S} \/ {NAT} by TARSKI:def 2;
A4: ( S₁[k, NAT ] or S₁[k, the carrier of S] or S₁[k, SCM-Instr S] ) by Lm1;
the carrier of S in { the carrier of S} \/ {NAT} by A2, TARSKI:def 2;
hence ex b being Element of { the carrier of S} \/ {NAT} st S₁[k,b] by A3, A4; :: thesis: verum

end;

consider h being Function of SCM-Memory,({ the carrier of S} \/ {NAT}) such that
A5: for a being Element of SCM-Memory holds S₁[a,h . a] from FUNCT_2:sch 3(A1);
take h ; :: thesis: for k being Element of SCM-Memory holds
( ( k = NAT implies h . k = NAT ) & ( k in SCM-Data-Loc implies h . k = the carrier of S ) )
let k be Element of SCM-Memory ; :: thesis: ( ( k = NAT implies h . k = NAT ) & ( k in SCM-Data-Loc implies h . k = the carrier of S ) )
A6: S₁[k,h . k] by A5;
hence ( k = NAT implies h . k = NAT ) by AMI_2:20; :: thesis: ( k in SCM-Data-Loc implies h . k = the carrier of S )
thus ( k in SCM-Data-Loc implies h . k = the carrier of S ) by A6, AMI_2:20, AMI_2:21, XBOOLE_0:3; :: thesis: verum
thus verum by A6, AMI_2:21, XBOOLE_0:3; :: thesis: verum