defpred S₁[ set , set ] means ( ( $1 = NAT & $2 = NAT ) or ( $1 in SCM-Data-Loc & $2 = INT ) or ( $1 in NAT & $2 = SCM-Instr ) );

A1: now

let k be Element of SCM-Memory ; :: thesis: ex b being Element of {INT } \/ {SCM-Instr ,NAT } st S₁[k,b]
A2: {INT } \/ {SCM-Instr ,NAT } = {INT ,SCM-Instr ,NAT } by ENUMSET1:42;
then A3: NAT in {INT } \/ {SCM-Instr ,NAT } by ENUMSET1:def 1;
A4: ( S₁[k, NAT ] or S₁[k, INT ] or S₁[k, SCM-Instr ] ) by Lm1;
( INT in {INT } \/ {SCM-Instr ,NAT } & SCM-Instr in {INT } \/ {SCM-Instr ,NAT } ) by A2, ENUMSET1:def 1;
hence ex b being Element of {INT } \/ {SCM-Instr ,NAT } st S₁[k,b] by A3, A4; :: thesis: verum

end;

consider h being Function of SCM-Memory ,({INT } \/ {SCM-Instr ,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 = INT ) & ( k in NAT implies h . k = SCM-Instr ) )
let k be Element of SCM-Memory ; :: thesis: ( ( k = NAT implies h . k = NAT ) & ( k in SCM-Data-Loc implies h . k = INT ) & ( k in NAT implies h . k = SCM-Instr ) )
A6: S₁[k,h . k] by A5;
hence ( k = NAT implies h . k = NAT ) by Lm2; :: thesis: ( ( k in SCM-Data-Loc implies h . k = INT ) & ( k in NAT implies h . k = SCM-Instr ) )
thus ( k in SCM-Data-Loc implies h . k = INT ) by A6, Lm2, Lm3, XBOOLE_0:3; :: thesis: ( k in NAT implies h . k = SCM-Instr )
thus ( k in NAT implies h . k = SCM-Instr ) by A6, Lm3, XBOOLE_0:3; :: thesis: verum