reserve a, I for set,
  S for non empty non void ManySortedSign;
reserve A, M for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem
  for A being MSAlgebra over S for C1, C2 being MSEquivalence-like
  ManySortedRelation of A st C1 c= C2 for i being Element of S for x, y being
Element of (the Sorts of A).i st [x,y] in C1.i holds Class (C1,x) c= Class (C2,
  y) & (A is non-empty implies Class (C1,y) c= Class (C2,x))
proof
  let A be MSAlgebra over S, C1, C2 be MSEquivalence-like ManySortedRelation
  of A such that
A1: C1 c= C2;
  let i be Element of S, x, y be Element of (the Sorts of A).i such that
A2: [x,y] in C1.i;
  field(C1.i) = (the Sorts of A).i by ORDERS_1:12;
  then
A3: C1.i is_transitive_in (the Sorts of A).i by RELAT_2:def 16;
A4: C1.i c= C2.i by A1;
  thus Class (C1,x) c= Class (C2,y)
  proof
    let q be object;
    assume
A5: q in Class (C1,x);
    then [q,x] in C1.i by EQREL_1:19;
    then [q,y] in C1.i by A2,A3,A5,RELAT_2:def 8;
    hence thesis by A4,EQREL_1:19;
  end;
  assume A is non-empty;
  then reconsider B = A as non-empty MSAlgebra over S;
  field(C1.i) = (the Sorts of A).i by ORDERS_1:12;
  then C1.i is_symmetric_in (the Sorts of B).i by RELAT_2:def 11;
  then
A6: [y,x] in C1.i by A2,RELAT_2:def 3;
  let q be object such that
A7: q in Class (C1,y);
  [q,y] in C1.i by A7,EQREL_1:19;
  then [q,x] in C1.i by A3,A7,A6,RELAT_2:def 8;
  hence thesis by A4,EQREL_1:19;
end;
