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 Th8:
  u,v // y#u,y#v
proof
  set p=y#u,r=y#v;
A1: y+u = p+p & y+v = r+r by Def2;
  2*(r-p) = (1+1)*r - (1+1)*p by RLVECT_1:34
    .= (1*r+1*r)-(1+1)*p by RLVECT_1:def 6
    .= (1*r+1*r)-(1*p+1*p) by RLVECT_1:def 6
    .= (r+1*r)-(1*p+1*p) by RLVECT_1:def 8
    .= (r+r)-(1*p+1*p) by RLVECT_1:def 8
    .= (r+r)-(p+1*p) by RLVECT_1:def 8
    .= (y+v)-(y+u) by A1,RLVECT_1:def 8
    .= v+(y-(y+u)) by RLVECT_1:def 3
    .= v+((y-y)-u) by RLVECT_1:27
    .= v+(0.V-u) by RLVECT_1:15
    .= v-u by RLVECT_1:14
    .= 1*(v-u) by RLVECT_1:def 8;
  hence thesis by ANALOAF:def 1;
end;
