reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;

theorem Th6:
  (u#u1)#(v#v1)=(u#v)#(u1#v1)
proof
  set p=u#u1, q=v#v1, r=u#v, s=u1#v1;
  set x=p#q,y=r#s;
A1: now
    let w;
    w=1*w by RLVECT_1:def 8;
    then (w+w)+(w+w) = (1+1)*w + ((1*w)+(1*w)) by RLVECT_1:def 6
      .= (1+1)*w + (1+1)*w by RLVECT_1:def 6
      .= ((1+1)+(1+1))*w by RLVECT_1:def 6;
    hence (w+w)+(w+w) = 4*w;
  end;
  (x+x)+(x+x) = (x+x)+(p+q) by Def2
    .= (p+q)+(p+q) by Def2
    .= p+(q+(p+q)) by RLVECT_1:def 3
    .= p+(p+(q+q)) by RLVECT_1:def 3
    .= (p+p)+(q+q) by RLVECT_1:def 3
    .= (p+p)+(v+v1) by Def2
    .= (u+u1)+(v+v1) by Def2
    .= u+(u1+(v+v1)) by RLVECT_1:def 3
    .= u+(v+(v1+u1)) by RLVECT_1:def 3
    .= (u+v)+(v1+u1) by RLVECT_1:def 3
    .= (u+v)+(s+s) by Def2
    .= (r+r)+(s+s) by Def2
    .= r+(r+(s+s)) by RLVECT_1:def 3
    .= r+(s+(s+r)) by RLVECT_1:def 3
    .= (r+s)+(s+r) by RLVECT_1:def 3
    .= (y+y)+(r+s) by Def2
    .= (y+y)+(y+y) by Def2;
  then 4*x = (y+y)+(y+y) by A1
    .= 4*y by A1;
  hence thesis by RLVECT_1:36;
end;
