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
  X is DIAMOND iff the reduction of X is subcommutative
  proof
    set R = the reduction of X;
    set A = the carrier of X;
F0: field R c= A \/ A by RELSET_1:8;
    thus X is DIAMOND implies R is subcommutative
    proof assume
A1:   x <<01>> y implies x >>01<< y;
      let a,b,c be object;
      assume
A2:   [a,b] in R & [a,c] in R; then
      a in field R & b in field R & c in field R by RELAT_1:15; then
      reconsider x = a, y = b, z = c as Element of X by F0;
      x ==> y & x ==> z by A2; then
      x =01=> y & x =01=> z; then
      y <<01>> z;
      hence b,c are_convergent<=1_wrt R by A1,Ch14;
    end;
    assume
A3: for a,b,c being object st [a,b] in R & [a,c] in R
    holds b,c are_convergent<=1_wrt R;
    let x,y; given z such that
A4: x <=01= z & z =01=> y;
    per cases by A4;
    suppose
      x <== z & z ==> y;
      hence thesis by A3,Ch14;
    end;
    suppose
      x = z & z = y;
      hence thesis;
    end;
    suppose
      x <== z & z = y;
      hence thesis by Th17;
    end;
    suppose
      x = z & z ==> y;
      hence thesis by Th17;
    end;
  end;
