reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;

theorem Th50:
  for X1 being Subset of X, Y1 being Subset of Y st [:X1,Y1:] <>
  {} holds .:pr1(X,Y). [:X1,Y1:] = X1 & .:pr2(X,Y). [:X1,Y1:] = Y1
proof
  let X1 be Subset of X, Y1 be Subset of Y;
  assume
A1: [:X1,Y1:] <> {};
  thus .:pr1(X,Y). [:X1,Y1:] = pr1(X,Y).:[:X1,Y1:] by Th47
    .= X1 by A1,Th49;
  thus .:pr2(X,Y). [:X1,Y1:] = pr2(X,Y).:[:X1,Y1:] by Th48
    .= Y1 by A1,Th49;
end;
