reserve v for object;
reserve V,A for set;
reserve f for SCBinominativeFunction of V,A;
reserve d for TypeSCNominativeData of V,A;
reserve d1 for NonatomicND of V,A;
reserve a,b,c,z for Element of V;
reserve x,y for object;
reserve p,q,r,s for SCPartialNominativePredicate of V,A;

theorem
  (for d holds a is_a_value_on d) & (for d holds b is_a_value_on d) &
  a is_ext_real_on d & b is_ext_real_on d &
  d in dom(PP_not Equality(A,a,b)) & (PP_not Equality(A,a,b)).d = TRUE implies
  d in dom less(A,a,b) & less(A,a,b).d = TRUE or
  d in dom less(A,b,a) & less(A,b,a).d = TRUE
  proof
    set e = Equality(A,a,b);
    set E = Equality(A);
    set Da = denaming(V,A,a);
    set Db = denaming(V,A,b);
    set L = less(A);
    assume that
A1: (for d holds a is_a_value_on d) & (for d holds b is_a_value_on d) and
A2: a is_ext_real_on d & b is_ext_real_on d and
A3: d in dom PP_not e and
A4: (PP_not e).d = TRUE and
A5: not d in dom less(A,a,b) or less(A,a,b).d <> TRUE;
A6: a is_a_value_on d by A1;
A7: b is_a_value_on d by A1;
A8: dom <:Db,Da:> = dom Db /\ dom Da by FUNCT_3:def 7;
A9: dom PP_not e = dom e by PARTPR_1:def 2;
A10: dom e c= dom <:Da,Db:> by RELAT_1:25;
    then d in dom <:Da,Db:> by A3,A9;
    then
A11: d in dom <:Db,Da:> by A8,FUNCT_3:def 7;
A12: e.d = FALSE by A3,A4,A9,PARTPR_1:5;
A13: <:Da,Db:>.d = [Da.d,Db.d] by A3,A9,A10,FUNCT_3:def 7;
    then e.d = E.(Da.d,Db.d) by A3,A9,A10,FUNCT_1:13;
    then
A14: Da.d <> Db.d by A6,A12,Def9;
    reconsider x = Da.d, y = Db.d as ExtReal by A2;
A15: d in dom <:Da,Db:> by A3,A9,FUNCT_1:11;
A16: dom L = [:A,A:] by FUNCT_2:def 1;
    [Da.d,Db.d] in [:A,A:] by A6,A7,ZFMISC_1:def 2;
    then d in dom(L*<:Da,Db:>) by A15,A13,A16,FUNCT_1:11;
    then FALSE = (L*<:Da,Db:>).d by A5,PARTPR_1:3
    .= L.(Da.d,Db.d) by A3,A9,A10,A13,FUNCT_1:13;
    then not x less_pred y by A6,A7,Def12;
    then
A17: Db.d less_pred Da.d by A14,Th10;
    (L*<:Db,Da:>).d = L.(<:Db,Da:>.d) by A11,FUNCT_1:13
    .= L.(Db.d,Da.d) by A11,FUNCT_3:def 7
    .= TRUE by A6,A7,A17,Def12;
    hence thesis by A1,A8,A11,Th12;
  end;
