
theorem Th18: :: DCompl1:
for R being RelStr, x, y being Element of R,
    a, b being Element of ComplRelStr R
 st x = a & y = b & x <> y & x in the carrier of R & not a <= b holds x <= y
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 and
A3: x <> y and
A4: x in the carrier of R;
   set cR = the carrier of R, iR = the InternalRel of R;
   set CR = ComplRelStr R;
   set iCR = the InternalRel of CR;
A5: iCR = iR` \ id cR by NECKLACE:def 8;
A6: [x,y] in [:cR,cR:] by A4,ZFMISC_1:def 2;
   assume not a <= b;
    then A7: not [x,y] in iCR by A1,A2,ORDERS_2:def 5;
       not [x,y] in id cR by A3,RELAT_1:def 10;
    then not [x,y] in iR` by A5,A7,XBOOLE_0:def 5;
    then [x,y] in iR by A6,XBOOLE_0:def 5;
   hence x <= y by ORDERS_2:def 5;
end;
