
theorem Th45: :: Msubrel:
for n being Nat, R being NatRelStr of n, S being Subset of Mycielskian R
 st S = n holds R = subrelstr S
proof
 let n be Nat, R be NatRelStr of n, S be Subset of Mycielskian R such that
A1: S = n;
   set cR = the carrier of R, iR = the InternalRel of R;
   set sS = subrelstr S; set csS = the carrier of sS;
   set isS = the InternalRel of sS;
   set MR = Mycielskian R;
   set cMR = the carrier of MR, iMR = the InternalRel of MR;
A2: cR = n by Def7;
A3: csS = n by A1,YELLOW_0:def 15;
A4: iR = isS proof
    thus iR c= isS proof
      let z be object;
      assume A5: z in iR;
       then consider x, y being object such that
      A6: x in cR and
      A7: y in cR and
      A8: z = [x,y] by ZFMISC_1:def 2;
          cR c= cMR by Th37;
          then reconsider xMR = x, yMR = y as Element of MR by A6,A7;
          reconsider xsS = x, ysS = y as Element of sS by A6,A7,Def7,A3;
          iR c= iMR by Th39;
          then xMR <= yMR by A5,A8,ORDERS_2:def 5;
          then xsS <= ysS by A3,A2,A7,YELLOW_0:60;
      hence z in isS by A8,ORDERS_2:def 5;
    end;
    thus isS c= iR proof
      let z be object;
      assume A9: z in isS;
       then consider x, y being object such that
      A10: x in Segm n and
      A11: y in Segm n and
      A12: z = [x,y] by ZFMISC_1:def 2,A3;
          cR c= cMR by Th37;
          then reconsider xMR = x, yMR = y as Element of MR by A10,A11,A2;
          reconsider xsS = x, ysS = y as Element of sS by A10,A11,A3;
          xsS <= ysS by A9,A12,ORDERS_2:def 5;
          then xMR <= yMR by YELLOW_0:59;
          then z in iMR by A12,ORDERS_2:def 5;
      hence z in iR by A10,A11,A12,Th40;
    end;
   end;
 thus R = subrelstr S by Def7,A3,A4;
end;
