let IT1, IT2 be Element of R; :: thesis: ( IT1 in X & IT1 is_minimal_wrt X, the InternalRel of R & IT2 in X & IT2 is_minimal_wrt X, the InternalRel of R implies IT1 = IT2 )
assume that
A4: IT1 in X and
A5: IT1 is_minimal_wrt X, the InternalRel of R and
A6: IT2 in X and
A7: IT2 is_minimal_wrt X, the InternalRel of R ; :: thesis: IT1 = IT2
set IR = the InternalRel of R;
A8: ( IT1 <= IT2 or IT2 <= IT1 ) by WAYBEL_0:def 29;

per cases ( [IT1,IT2] in the InternalRel of R or [IT2,IT1] in the InternalRel of R ) by A8, ORDERS_2:def 5;

suppose [IT1,IT2] in the InternalRel of R ; :: thesis: IT1 = IT2

hence IT1 = IT2 by A4, A7, WAYBEL_4:def 25; :: thesis: verum

end;

suppose [IT2,IT1] in the InternalRel of R ; :: thesis: IT1 = IT2

hence IT1 = IT2 by A5, A6, WAYBEL_4:def 25; :: thesis: verum

end;

end;