reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;

theorem Th12:
  a^2 + b^2 = g^2 iff |[a,b]| in circle(0,0,g)
  proof
    hereby
      assume
A1:   a^2 + b^2 = g^2;
      reconsider p = |[a,b]| as Point of TOP-REAL 2;
      |. p - |[0,0]| .| = g
      proof
A2:     |. p - |[0,0]| .|^2 = |. |[a - 0,b - 0]| .|^2 by EUCLID:62
                         .= g^2 by A1,TOPGEN_5:9;
        -g < 0;
        hence thesis by A2,SQUARE_1:40;
      end;
      hence |[a,b]| in circle(0,0,g);
    end;
    assume
A3: |[a,b]| in  circle(0,0,g);
    thus a^2 + b^2 = |. |[a - 0,b - 0]| .|^2 by TOPGEN_5:9
                  .= |. |[a,b]| - |[0,0]| .|^2 by EUCLID:62
                  .= g^2 by A3,TOPREAL9:43;
  end;
