reserve A,O for non empty set,
  R for Order of A,
  Ol for Equivalence_Relation of O,
  f for Function of O,A*,
  g for Function of O,A;
reserve S for OverloadedRSSign;
reserve S0 for non empty non void ManySortedSign;
reserve S for non empty Poset;
reserve s1,s2 for Element of S;
reserve w1,w2 for Element of (the carrier of S)*;

theorem Th7:
  S is discrete & w1 <= w2 implies w1 = w2
proof
  assume that
A1: S is discrete and
A2: w1 <= w2;
  reconsider S1 = S as discrete non empty Poset by A1;
  len w1 = len w2 by A2;
  then
A3: dom w1 = dom w2 by FINSEQ_3:29;
  for i being object st i in dom w1 holds w1.i = w2.i
  proof
    let i be object such that
A4: i in dom w1;
    reconsider s3 = w1.i, s4 = w2.i as Element of S by A3,A4,PARTFUN1:4;
    reconsider s5 = s3, s6 = s4 as Element of S1;
    s3 <= s4 by A2,A4;
    then s5 = s6 by ORDERS_3:1;
    hence thesis;
  end;
  hence thesis by A3,FUNCT_1:2;
end;
