defpred S₁[ set , set ] means ( ( $1 = NAT & $2 = 0 ) or ( $1 in SCM-Data-Loc & $2 = 1 ) );

A1: now :: thesis: for k being Element of SCM-Memory ex b being Element of Segm 2 st S₁[k,b]

let k be Element of SCM-Memory ; :: thesis: ex b being Element of Segm 2 st S₁[k,b]
A2: {0} \/ {1} = {0,1} by ENUMSET1:1;
then A3: 0 in {1} \/ {0} by TARSKI:def 2;
A4: ( S₁[k, 0 ] or S₁[k,1] ) by Lm1;
1 in {1} \/ {0} by A2, TARSKI:def 2;
hence ex b being Element of Segm 2 st S₁[k,b] by A2, A3, A4, CARD_1:50; :: thesis: verum

end;

consider h being Function of SCM-Memory,(Segm 2) 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 = 0 ) & ( k in SCM-Data-Loc implies h . k = 1 ) )
let k be Element of SCM-Memory ; :: thesis: ( ( k = NAT implies h . k = 0 ) & ( k in SCM-Data-Loc implies h . k = 1 ) )
thus ( k = NAT implies h . k = 0 ) by A5, Lm2; :: thesis: ( k in SCM-Data-Loc implies h . k = 1 )
thus ( k in SCM-Data-Loc implies h . k = 1 ) by A5, Lm2; :: thesis: verum
thus verum ; :: thesis: verum