theorem ThL7:
  c1 <> c2 implies (root-tree c1,c2)<-d = root-tree c1
  proof
    assume
A0: c1 <> c2;
AA: dom ((root-tree c1,c2)<-d) = dom root-tree c1
    proof
      let x be FinSequence of NAT;
      hereby assume
        x in dom((root-tree c1,c2)<-d);
        then reconsider p = x as Element of dom((root-tree c1,c2)<-d);
        per cases by TREES_4:def 7;
        suppose p in dom root-tree c1;
          hence x in dom root-tree c1;
        end;
        suppose
          ex q being Node of root-tree c1, r being Node of d st
          q in Leaves dom root-tree c1 & (root-tree c1).q = c2 & p = q^r;
          then consider q being Node of root-tree c1, r being Node of d such
          that
A2:       q in Leaves dom root-tree c1 & (root-tree c1).q = c2 & p = q^r;
          thus x in dom root-tree c1 by A2,A0;
        end;
      end;
      assume x in dom root-tree c1;
      hence x in dom((root-tree c1,c2)<-d) by TREES_4:def 7;
    end;
    now let x be object;
      assume x in dom root-tree c1;
      then reconsider p = x as Node of root-tree c1;
      (root-tree c1).p = c1;
      hence ((root-tree c1,c2)<-d).x = (root-tree c1).x by A0,TREES_4:def 7;
    end;
    hence thesis by AA,FUNCT_1:2;
  end;
