reserve IIG for Circuit-like non void non empty ManySortedSign;

theorem
  for IIG for SCS being non-empty Circuit of IIG, v being Vertex of IIG,
  e being Element of (the Sorts of SCS).v st v in SortsWithConstants IIG & e in
  Constants (SCS, v) holds (Set-Constants SCS).v = e
proof
  let IIG;
  let SCS be non-empty Circuit of IIG, v be Vertex of IIG, e be Element of (
  the Sorts of SCS).v;
  assume that
A1: v in SortsWithConstants IIG and
A2: e in Constants (SCS, v);
A3: ex A being non empty set st A =(the Sorts of SCS).v & Constants (SCS, v)
= { a where a is Element of A : ex o be OperSymbol of IIG st (the Arity of IIG)
.o = {} & (the ResultSort of IIG).o = v & a in rng Den(o,SCS) } by
MSUALG_2:def 3;
  then ex a being Element of (the Sorts of SCS).v st a = e & ex o being
OperSymbol of IIG st (the Arity of IIG).o = {} & (the ResultSort of IIG ).o = v
  & a in rng Den(o,SCS) by A2;
  then consider o being OperSymbol of IIG such that
A4: (the Arity of IIG).o = {} and
A5: (the ResultSort of IIG).o = v and
A6: e in rng Den(o,SCS);
A7: {} = <*>the carrier of IIG;
  v in dom Set-Constants SCS by A1,PARTFUN1:def 2;
  then (Set-Constants SCS).v in Constants (SCS, v) by Def1;
  then
  ex a being Element of (the Sorts of SCS).v st a = ( Set-Constants SCS).v
  & ex o being OperSymbol of IIG st (the Arity of IIG).o = {} & (the ResultSort
  of IIG).o = v & a in rng Den(o,SCS) by A3;
  then consider o1 being OperSymbol of IIG such that
  (the Arity of IIG).o1 = {} and
A8: (the ResultSort of IIG).o1 = v and
A9: (Set-Constants SCS).v in rng Den(o1,SCS);
A10: ex d1 being object
st d1 in dom Den(o1, SCS) & ( Set-Constants SCS).v = Den
  (o1, SCS).d1 by A9,FUNCT_1:def 3;
  the_result_sort_of o = (the ResultSort of IIG).o & the_result_sort_of
  o1 = ( the ResultSort of IIG).o1 by MSUALG_1:def 2;
  then
A11: o = o1 by A5,A8,MSAFREE2:def 6;
  consider d being object such that
A12: d in dom Den(o, SCS) and
A13: e = Den(o, SCS).d by A6,FUNCT_1:def 3;
A14: dom the Arity of IIG = the carrier' of IIG by FUNCT_2:def 1;
A15: dom Den (o, SCS) = Args (o, SCS) by FUNCT_2:def 1
    .= ((the Sorts of SCS)# * the Arity of IIG).o by MSUALG_1:def 4
    .= (the Sorts of SCS)#.((the Arity of IIG).o) by A14,FUNCT_1:13
    .= {{}} by A4,A7,PRE_CIRC:2;
  then d = {} by A12,TARSKI:def 1;
  hence thesis by A11,A15,A13,A10,TARSKI:def 1;
end;
