reserve X for ARS, a,b,c,u,v,w,x,y,z for Element of X;
reserve i,j,k for Element of ARS_01;
reserve l,m,n for Element of ARS_02;
reserve A for set;

theorem LemA:
  X is DIAMOND & x <=*= z & z =01=> y implies
  ex u st x =01=> u & u <=*= y
  proof
    assume
A1: for x,y st x <<01>> y holds x >>01<< y;
    assume
A2: x <=*= z;
    assume
A3: z =01=> y;
    defpred P[Element of X] means ex u st $1 =01=> u & u <=*= y;
A4: for u,v st u ==> v & P[u] holds P[v]
    proof
      let u,v;
      assume u ==> v; then
B1:   u =01=> v;
      given w such that
B2:   u =01=> w & w <=*= y;
      v <<01>> w by B1,B2; then
      v >>01<< w by A1; then
      consider u such that
B3:   v =01=> u & u <=01= w;
      thus P[v] by B2,B3,Lem11;
    end;
A5: for u,v st u =*=> v & P[u] holds P[v] from Star(A4);
    thus thesis by A5,A2,A3;
  end;
