reserve A for set,
  C for non empty set,
  B for Subset of A,
  x for Element of A,
  f,g for Function of A,C;
reserve B for Element of Fin A;
reserve L for non empty LattStr,
  a,b,c for Element of L;
reserve L for Lattice;
reserve a,b,c,u,v for Element of L;
reserve A for non empty set,
  x for Element of A,
  B for Element of Fin A,
  f,g for Function of A, the carrier of L;
reserve L for 0_Lattice,
  f,g for Function of A, the carrier of L,
  u for Element of L;
reserve L for 1_Lattice,
  f,g for Function of A, the carrier of L,
  u for Element of L;

theorem
  f|B = g|B implies FinMeet(B,f) = FinMeet(B,g)
proof
  assume
A1: f|B = g|B;
  reconsider f9 = f, g9 = g as Function of A, the carrier of L.:;
A2: FinMeet(B,g) = FinJoin(B,g9);
  L.: is 0_Lattice & FinMeet(B,f) = FinJoin(B,f9) by Th49;
  hence thesis by A1,A2,Th53;
end;
