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 Th10:
  2*(u#v-u) = v-u & 2*(v-u#v) = v-u
proof
  set p=u#v;
A1: 2-1 =1;
A2: 2*(v-p) = 2*v-(1+1)*p by RLVECT_1:34
    .= 2*v-(1*p+1*p) by RLVECT_1:def 6
    .= 2*v - (1*p+p) by RLVECT_1:def 8
    .= 2*v - (p+p) by RLVECT_1:def 8
    .= 2*v - (u+v) by Def2
    .= (2*v-v)-u by RLVECT_1:27
    .= (2*v-1*v)-u by RLVECT_1:def 8
    .= 1*v - u by A1,RLVECT_1:35
    .= v-u by RLVECT_1:def 8;
A3: 1-2 = -1;
  2*(p-u) = (1+1)*p - 2*u by RLVECT_1:34
    .= (1*p+1*p) - 2*u by RLVECT_1:def 6
    .= (p+1*p) - 2*u by RLVECT_1:def 8
    .= (p+p) - 2*u by RLVECT_1:def 8
    .= (u+v) - 2*u by Def2
    .=v+(u-2*u) by RLVECT_1:def 3
    .= v+(1*u-2*u) by RLVECT_1:def 8
    .= v+(-1)*u by A3,RLVECT_1:35
    .= v-u by RLVECT_1:16;
  hence thesis by A2;
end;
