thus not SCMPDS is empty by STRUCT_0:def 1; :: thesis: SCMPDS is standard-ins
consider X being non empty set such that
A1: proj2 SCMPDS-Instr c= X * by FINSEQ_1:85;
take X ; :: according to COMPOS_1:def 4 :: thesis: the Instructions of SCMPDS c= [:NAT,(NAT *),(X *):]
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in the Instructions of SCMPDS or x in [:NAT,(NAT *),(X *):] )
assume A2: x in the Instructions of SCMPDS ; :: thesis: x in [:NAT,(NAT *),(X *):]
A3: {} in NAT * by FINSEQ_1:49;

per cases ( x in (( { [0,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) \/ { [I,{},<*v,c*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ) \/ { [I,{},<*v,c1,c2*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {4,5,6,7,8} } or x in { [I,{},<*v1,v2,c1,c2*>] where I is Element of Segm 14, v1, v2 is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {9,10,11,12,13} } ) by A2, XBOOLE_0:def 3;

suppose A4: x in (( { [0,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) \/ { [I,{},<*v,c*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ) \/ { [I,{},<*v,c1,c2*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {4,5,6,7,8} } ; :: thesis: x in [:NAT,(NAT *),(X *):]

per cases ( x in ( { [0,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) \/ { [I,{},<*v,c*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } or x in { [I,{},<*v,c1,c2*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {4,5,6,7,8} } ) by A4, XBOOLE_0:def 3;

suppose A5: x in ( { [0,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) \/ { [I,{},<*v,c*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ; :: thesis: x in [:NAT,(NAT *),(X *):]

per cases ( x in { [0,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } or x in { [I,{},<*v,c*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ) by A5, XBOOLE_0:def 3;

suppose A6: x in { [0,{},<*l*>] where l is Element of INT : verum } \/ { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ; :: thesis: x in [:NAT,(NAT *),(X *):]

per cases ( x in { [0,{},<*l*>] where l is Element of INT : verum } or x in { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ) by A6, XBOOLE_0:def 3;

suppose x in { [0,{},<*l*>] where l is Element of INT : verum } ; :: thesis: x in [:NAT,(NAT *),(X *):]

then consider l being Element of INT such that
A7: x = [0,{},<*l*>] ;
<*l*> in proj2 SCMPDS-Instr by A2, A7, RELAT_1:def 5;
then <*l*> in X * by A1;
hence x in [:NAT,(NAT *),(X *):] by A7, A3, MCART_1:69; :: thesis: verum

end;

suppose x in { [1,{},<*sp*>] where sp is Element of SCM-Data-Loc : verum } ; :: thesis: x in [:NAT,(NAT *),(X *):]

then consider sp being Element of SCM-Data-Loc such that
A8: x = [1,{},<*sp*>] ;
<*sp*> in proj2 SCMPDS-Instr by A2, A8, RELAT_1:def 5;
then <*sp*> in X * by A1;
hence x in [:NAT,(NAT *),(X *):] by A8, A3, MCART_1:69; :: thesis: verum

end;

end;

end;

suppose x in { [I,{},<*v,c*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c is Element of INT : I in {2,3} } ; :: thesis: x in [:NAT,(NAT *),(X *):]

then consider I being Element of Segm 14, v being Element of SCM-Data-Loc , c being Element of INT such that
A9: ( x = [I,{},<*v,c*>] & I in {2,3} ) ;
<*v,c*> in proj2 SCMPDS-Instr by A2, A9, RELAT_1:def 5;
then <*v,c*> in X * by A1;
hence x in [:NAT,(NAT *),(X *):] by A9, A3, MCART_1:69; :: thesis: verum

end;

end;

end;

suppose x in { [I,{},<*v,c1,c2*>] where I is Element of Segm 14, v is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {4,5,6,7,8} } ; :: thesis: x in [:NAT,(NAT *),(X *):]

then consider I being Element of Segm 14, v being Element of SCM-Data-Loc , c1, c2 being Element of INT such that
A10: ( x = [I,{},<*v,c1,c2*>] & I in {4,5,6,7,8} ) ;
<*v,c1,c2*> in proj2 SCMPDS-Instr by A2, A10, RELAT_1:def 5;
then <*v,c1,c2*> in X * by A1;
hence x in [:NAT,(NAT *),(X *):] by A10, A3, MCART_1:69; :: thesis: verum

end;

end;

end;

suppose x in { [I,{},<*v1,v2,c1,c2*>] where I is Element of Segm 14, v1, v2 is Element of SCM-Data-Loc , c1, c2 is Element of INT : I in {9,10,11,12,13} } ; :: thesis: x in [:NAT,(NAT *),(X *):]

then consider I being Element of Segm 14, v1, v2 being Element of SCM-Data-Loc , c1, c2 being Element of INT such that
A11: ( x = [I,{},<*v1,v2,c1,c2*>] & I in {9,10,11,12,13} ) ;
<*v1,v2,c1,c2*> in proj2 SCMPDS-Instr by A2, A11, RELAT_1:def 5;
then <*v1,v2,c1,c2*> in X * by A1;
hence x in [:NAT,(NAT *),(X *):] by A11, A3, MCART_1:69; :: thesis: verum

end;

end;