
theorem
for F being Field
for S being Subset of F st S * S c= S & S \/ (-S) = the carrier of F
holds S /\ (-S) = {0.F} iff not -1.F in S
proof
let F be Field, S be Subset of F;
assume AS: S * S c= S & S \/ (-S) = the carrier of F;
SQ F c= S
proof let o be object;
  assume o in SQ F;
  then consider a being Element of F such that A: a = o & a is square;
  consider b being Element of F such that B: b^2 = a by A;
  per cases by AS,XBOOLE_0:def 3;
  suppose b in S;
    then b * b in S * S;
    hence o in S by AS,A,B;
    end;
  suppose b in -S;
    then -b in -(-S);
    then (-b) * (-b) in S * S;
    then (-b) * (-b) in S by AS;
    hence o in S by A,B,VECTSP_1:10;
    end;
  end;
hence thesis by X1,AS;
end;
