reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;
reserve A,B for Ordinal;
reserve n,k for Nat;

theorem Th104:
 for I being set, f being non-empty ManySortedSet of I
 for s being f-compatible ManySortedSet of I
  holds s in product f
 proof let I be set, f be non-empty ManySortedSet of I;
  let s be f-compatible ManySortedSet of I;
A1: dom s = I by PARTFUN1:def 2 .= dom f by PARTFUN1:def 2;
   then for x being object st x in dom f holds s.x in f.x by FUNCT_1:def 14;
  hence s in product f by A1,Th9;
 end;
