reserve n,n1,n2,m for Nat;
reserve r,g1,g2,g,g9 for Complex;
reserve R,R2 for Real;
reserve s,s9,s1 for Complex_Sequence;

theorem Th1:
  g<>0c & r<>0c implies |.g"-r".|=|.g-r.|/(|.g.|*|.r.|)
proof
  assume
A1: g<>0c & r<>0c;
  thus |.g"-r".|=|.1r/g-r".| by COMPLEX1:def 4,XCMPLX_1:215
    .=|.1r/g-1r/r.| by COMPLEX1:def 4,XCMPLX_1:215
    .=|.1r/g+-1r/r.|
    .=|.1r/g+-1r*r".| by XCMPLX_0:def 9
    .=|.1r/g+(-1r)*r".|
    .=|.1r/g+(-1r)/r.| by XCMPLX_0:def 9
    .=|.(1r*r+(-1r)*g)/(r*g).| by A1,XCMPLX_1:116
    .=|.r-g.|/|.g*r.| by COMPLEX1:67,def 4
    .=|.-(g-r).|/(|.g.|*|.r.|) by COMPLEX1:65
    .=|.g-r.|/(|.g.|*|.r.|) by COMPLEX1:52;
end;
