reserve X for BCI-algebra;
reserve n for Nat;
reserve x,y for Element of X;
reserve a,b for Element of AtomSet(X);
reserve m,n for Nat;
reserve i,j for Integer;

theorem Th1:
  a\(x\b) = b\(x\a)
proof
  b in AtomSet(X);
  then
A1: ex y1 being Element of X st b=y1 & y1 is atom;
  (x\(x\b))\b=0.X by BCIALG_1:1;
  then (b\(x\a))\(a\(x\b))= ((x\(x\b))\(x\a))\(a\(x\b)) by A1;
  then (b\(x\a))\(a\(x\b))= ((x\(x\a))\(x\b))\(a\(x\b))by BCIALG_1:7;
  then (b\(x\a))\(a\(x\b))= ((x\(x\a))\(x\b))\(a\(x\b))\0.X by BCIALG_1:2;
  then (b\(x\a))\(a\(x\b)) =((x\(x\a))\(x\b))\(a\(x\b))\((x\(x\a))\a) by
BCIALG_1:1;
  then
A2: b\(x\a)\(a\(x\b))=0.X by BCIALG_1:def 3;
  a\(x\b)<=(b\(x\a)) by BCIALG_1:26;
  then a\(x\b)\(b\(x\a)) = 0.X;
  hence thesis by A2,BCIALG_1:def 7;
end;
