reserve x,y,z for set;
reserve I,J,K for Element of Segm 9,
  i,a,a1,a2 for Nat,
  b,b1,b2,c,c1 for Element of SCM-Data-Loc;

theorem Th8:
 SCM-Instr c= [:NAT,NAT*,proj2 SCM-Instr:]
proof
 let x be object;
 assume
A1:  x in SCM-Instr;
 per cases by A1,XBOOLE_0:def 3;
 suppose
A2: x in {[SCM-Halt,{},{}] }
  \/ { [J,<*a2*>,{}] : J = 6 } \/ { [K,<*a1*>,<*b1*>] : K in { 7,8 } };
 per cases by A2,XBOOLE_0:def 3;
 suppose
A3: x in {[SCM-Halt,{},{}] } \/ { [J,<*a2*>,{}] : J = 6 };
 per cases by A3,XBOOLE_0:def 3;
 suppose x in {[SCM-Halt,{},{}] };
  then
A4: x = [SCM-Halt,{},{}] by TARSKI:def 1;
  then SCM-Halt in NAT & {} in NAT* &
   {} in proj2 SCM-Instr by A1,FINSEQ_1:49,XTUPLE_0:def 13;
 hence x in [:NAT,NAT*,proj2 SCM-Instr:] by A4,DOMAIN_1:3;
 end;
 suppose x in { [J,<*a2*>,{}] : J = 6 };
  then consider J,a such that
A5: x = [J,<*a*>,{}] & J = 6;
  J in NAT & <*a*> in NAT* & {} in proj2 SCM-Instr
   by A1,A5,FUNCT_7:18,XTUPLE_0:def 13,ORDINAL1:def 12;
 hence x in [:NAT,NAT*,proj2 SCM-Instr:] by A5,DOMAIN_1:3;
 end;
 end;
 suppose x in { [K,<*a1*>,<*b1*>] : K in { 7,8 }};
  then consider K,a1,b1 such that
A6: x = [K,<*a1*>,<*b1*>] & K in { 7,8 };
  K in NAT & <*a1*> in NAT* &
  <*b1*> in proj2 SCM-Instr
   by A1,A6,FUNCT_7:18,XTUPLE_0:def 13,ORDINAL1:def 12;
 hence x in [:NAT,NAT*,proj2 SCM-Instr:] by A6,DOMAIN_1:3;
 end;
 end;
 suppose x in { [I,{},<*b,c*>] : I in { 1,2,3,4,5} };
  then consider I,b,c such that
A7: x = [I,{},<*b,c*>] & I in { 1,2,3,4,5};
  I in NAT & {} in NAT* &
   <*b,c*> in proj2 SCM-Instr by A1,A7,FINSEQ_1:49,XTUPLE_0:def 13;
 hence x in [:NAT,NAT*,proj2 SCM-Instr:] by A7,DOMAIN_1:3;
 end;
end;
