reserve i, j, k for Nat,
  I for Element of Segm 8,
  i1, i2 for Nat,
  d1, d2, d3, d4 for Element of SCM-Data-Loc,
  S for non empty 1-sorted;
reserve G for non empty 1-sorted;

theorem Th5:
 for S being non empty 1-sorted
  holds  SCM-Instr S c= [:NAT,NAT*,proj2 SCM-Instr S:]
proof let S be non empty 1-sorted;
 set X=proj2 SCM-Instr S;
 let u be object;
 assume
A1: u in SCM-Instr S;
A2: {} in NAT* by FINSEQ_1:49;
 per cases by A1,XBOOLE_0:def 3;
 suppose
A3:  u in { [0,{},{}] } \/ { [I,{},<*a,b*>] where I is Element of Segm 8
, a, b is Element of SCM-Data-Loc: I in { 1,2,3,4 } }
 \/ the set of all  [6,<*i*>,{}] where i is Nat \/
  the set of all  [7,<*i*>,<*a*>] where i is Nat,
  a is Element of SCM-Data-Loc;
  per cases by A3,XBOOLE_0:def 3;
  suppose
A4:  u in{ [0,{},{}] } \/ { [I,{},<*a,b*>] where I is Element of Segm 8
, a, b is Element of SCM-Data-Loc: I in { 1,2,3,4 } }
 \/ the set of all  [6,<*i*>,{}] where i is Nat;
  per cases by A4,XBOOLE_0:def 3;
  suppose
A5:  u in { [0,{},{}] } \/ { [I,{},<*a,b*>] where I is Element of Segm 8
, a, b is Element of SCM-Data-Loc: I in { 1,2,3,4 } };
  per cases by A5,XBOOLE_0:def 3;
  suppose u in { [0,{},{}] };
   then
A6:  u = [0,{},{}] by TARSKI:def 1;
   then 0 in NAT & {} in proj2 SCM-Instr S by A1,XTUPLE_0:def 13;
  hence u in [:NAT,NAT*,X:] by A6,A2,MCART_1:69;
  end;
  suppose u in { [I,{},<*a,b*>] where I is Element of Segm 8
, a, b is Element of SCM-Data-Loc: I in { 1,2,3,4 }};
   then consider I being Element of Segm 8, a, b being Element of SCM-Data-Loc
   such that
A7: u = [I,{},<*a,b*>] & I in { 1,2,3,4 };
   I in NAT & <*a,b*> in proj2 SCM-Instr S by A7,A1,XTUPLE_0:def 13;
  hence u in [:NAT,NAT*,X:] by A7,A2,MCART_1:69;
  end;
  end;
  suppose u in the set of all  [6,<*i*>,{}] where i is Nat;
   then consider i being Nat such that
A8:  u = [6,<*i*>,{}];
   i in NAT by ORDINAL1:def 12;
  then
A9:  <*i*> in NAT* by FUNCT_7:18;
   6 in NAT & {} in proj2 SCM-Instr S by A8,A1,XTUPLE_0:def 13;
  hence u in [:NAT,NAT*,X:] by A8,A9,MCART_1:69;
  end;
  end;
  suppose u in the set of all  [7,<*i*>,<*a*>] where i is Nat,
  a is Element of SCM-Data-Loc;
   then consider i being Nat, a being Element of SCM-Data-Loc
   such that
A10:  u = [7,<*i*>,<*a*>];
   i in NAT by ORDINAL1:def 12;
   then
A11:  <*i*> in NAT* by FUNCT_7:18;
   7 in NAT & <*a*> in proj2 SCM-Instr S by A10,A1,XTUPLE_0:def 13;
  hence u in [:NAT,NAT*,X:] by A10,A11,MCART_1:69;
  end;
 end;
 suppose u in the set of all  [5,{},<*a,r*>] where a is
  Element of SCM-Data-Loc, r is Element of S;
  then consider a being Element of SCM-Data-Loc, r being Element of S such that
A12: u = [5,{},<*a,r*>];
   5 in NAT & <*a,r*> in proj2 SCM-Instr S by A12,A1,XTUPLE_0:def 13;
  hence u in [:NAT,NAT*,X:] by A12,A2,MCART_1:69;
 end;
end;
