reserve S for OrderSortedSign;
reserve S for OrderSortedSign,
  X for ManySortedSet of S,
  o for OperSymbol of S ,
  b for Element of ([:the carrier' of S,{the carrier of S}:] \/ Union (coprod X
  ))*;
reserve x for set;

theorem Th23:
  for S be locally_directed OrderSortedSign, X be non-empty
  ManySortedSet of S, x,y,s being object
   st [x,s] in (PTClasses X).y holds x in TS
  DTConOSA(X) & y in TS DTConOSA(X) & s in the carrier of S
proof
  let S be locally_directed OrderSortedSign, X be non-empty ManySortedSet of S;
  set D = DTConOSA(X), F = PTClasses X;
A1: rng F c= bool [:TS(D), the carrier of S:] by RELAT_1:def 19;
  let x,y,s being object such that
A2: [x,s] in (PTClasses X).y;
A3: y in TS D
  proof
    assume not y in TS D;
    then not y in dom F;
    hence contradiction by A2,FUNCT_1:def 2;
  end;
  dom F = TS(D) by FUNCT_2:def 1;
  then F.y in rng F by A3,FUNCT_1:3;
  hence thesis by A2,A1,A3,ZFMISC_1:87;
end;
