
theorem Th17: :: DCompl:
for R being RelStr, x, y being Element of R,
    a, b being Element of ComplRelStr R
 st x = a & y = b & x <= y holds not a <= b
proof
 let R be RelStr, x, y be Element of R,
     a, b be Element of ComplRelStr R such that
A1: x = a and
A2: y = b;
   set cR = the carrier of R, iR = the InternalRel of R;
   set CR = ComplRelStr R;
   set iCR = the InternalRel of CR;
A3: iCR = iR` \ id cR by NECKLACE:def 8;
     assume x <= y;
      then [x,y] in iR by ORDERS_2:def 5;
      then not [x,y] in iR` by XBOOLE_0:def 5;
      then not [x,y] in iCR by A3,XBOOLE_0:def 5;
     hence not a <= b by A1,A2,ORDERS_2:def 5;
end;
