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 Th24:
  for S be locally_directed OrderSortedSign, X be non-empty
ManySortedSet of S, C be Component of S, x,y being object
holds [x,y] in CompClass
  (PTCongruence X,C) iff ex s1 being Element of S st s1 in C & [x,s1] in (
  PTClasses X).y
proof
  let S be locally_directed OrderSortedSign, X be non-empty ManySortedSet of S
  , C be Component of S, x,y being object;
  hereby
    assume [x,y] in CompClass(PTCongruence X,C);
    then consider s1 being Element of S such that
A1: s1 in C and
A2: [x,y] in (PTCongruence X).s1 by OSALG_4:def 9;
A3: [x,y] in {[x1,y1] where x1,y1 is Element of TS(DTConOSA(X)): [x1,s1]
    in (PTClasses X).y1} by A2,Def22;
    take s1;
    consider x1,y1 being Element of TS(DTConOSA(X)) such that
A4: [x,y] = [x1,y1] and
A5: [x1,s1] in (PTClasses X).y1 by A3;
    x = x1 by A4,XTUPLE_0:1;
    hence s1 in C & [x,s1] in (PTClasses X).y by A1,A4,A5,XTUPLE_0:1;
  end;
  given s1 being Element of S such that
A6: s1 in C and
A7: [x,s1] in (PTClasses X).y;
  reconsider x2 = x, y2 = y as Element of TS(DTConOSA(X)) by A7,Th23;
  [x2,y2] in {[x1,y1] where x1,y1 is Element of TS(DTConOSA(X)): [x1,s1]
  in (PTClasses X).y1} by A7;
  then [x2,y2] in (PTCongruence(X)).s1 by Def22;
  hence thesis by A6,OSALG_4:def 9;
end;
