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;
reserve X,Y,Z for non empty set;

theorem
  for F being Function of X,Y, x being Element of X,
      z being Element of Z holds [:F,id Z:].(x,z) = [F.x,z]
proof
  let F be Function of X,Y, x be Element of X, z be Element of Z;
  thus [:F,id Z:].(x,z) = [F.x, (id Z).z] by FUNCT_3:75
    .= [F.x,z];
end;
