
theorem
{ 1, sqrt 2 } is Basis of VecSp(FAdj(F_Rat,{2-Root(2)}),F_Rat)
proof
set F = F_Rat;
B: now let o be object;
   assume o in Base 2-Root(2); then
   consider n being Element of NAT such that
   B1: o = 2-Root(2)|^n & n < deg MinPoly(2-Root(2),F);
   n < 1 + 1 by LL,B1,mp; then
   n <= 1 by NAT_1:13; then
   per cases by NAT_1:25;
   suppose n = 0;
    then o = 1_F_Real by B1,BINOM:8;
    hence o in {1, sqrt 2} by TARSKI:def 2;
    end;
   suppose n = 1;
    then o = 2-Root(2) by B1,BINOM:8;
    hence o in {1, sqrt 2} by TARSKI:def 2;
    end;
   end;
now let o be object;
  assume o in {1, sqrt 2}; then
  per cases by TARSKI:def 2;
  suppose o = 1;
    then o = 1_F_Real
          .= 2-Root(2)|^0 by BINOM:8;
    hence o in Base 2-Root(2) by LL,mp;
    end;
  suppose o = sqrt 2;
    then o = 2-Root(2)|^1 by BINOM:8;
    hence o in Base 2-Root(2) by LL,mp;
    end;
  end;
then Base 2-Root(2) = {1, sqrt 2} by B,TARSKI:2;
hence thesis by FIELD_6:65;
end;
