
theorem Th64:
  for V,W be VectSp of F_Complex, f be sesquilinear-Form of V,W
  holds leftker Q*Form(f) = leftker (RQ*Form(LQForm(f))) & rightker Q*Form(f) =
rightker (RQ*Form(LQForm(f))) & leftker Q*Form(f) = leftker (LQForm(RQ*Form(f))
  ) & rightker Q*Form(f) = rightker (LQForm(RQ*Form(f)))
proof
  set K =F_Complex;
  let V,W be VectSp of K, f be sesquilinear-Form of V,W;
  set vq = VectQuot(V,LKer(f)), wq = VectQuot(W,RKer(f*')), wqr = VectQuot(W,
  RKer(LQForm(f))*'), vql = VectQuot(V,LKer(RQ*Form(f)));
  set rlf = RQ*Form (LQForm(f)) , qf = Q*Form(f), lrf = LQForm (RQ*Form(f));
  thus leftker qf c= leftker rlf
  proof
    let x be object;
    assume x in leftker qf;
    then consider vv be Vector of vq such that
A1: x=vv and
A2: for ww be Vector of wq holds qf.(vv,ww)=0.K;
    now
      let ww be Vector of wqr;
      reconsider w = ww as Vector of wq by Th61;
      thus rlf.(vv,ww) = qf.(vv,w) by Th63
        .= 0.K by A2;
    end;
    hence thesis by A1;
  end;
  thus leftker rlf c= leftker qf
  proof
    let x be object;
    assume x in leftker rlf;
    then consider vv be Vector of vq such that
A3: x=vv and
A4: for ww be Vector of wqr holds rlf.(vv,ww)=0.K;
    now
      let ww be Vector of wq;
      reconsider w = ww as Vector of wqr by Th61;
      thus qf.(vv,ww) = rlf.(vv,w) by Th63
        .= 0.K by A4;
    end;
    hence thesis by A3;
  end;
  thus rightker qf c= rightker rlf
  proof
    let x be object;
    assume x in rightker qf;
    then consider ww be Vector of wq such that
A5: x=ww and
A6: for vv be Vector of vq holds qf.(vv,ww)=0.K;
    reconsider w = ww as Vector of wqr by Th61;
    now
      let vv be Vector of vq;
      thus rlf.(vv,w) = qf.(vv,ww) by Th63
        .= 0.K by A6;
    end;
    hence thesis by A5;
  end;
  thus rightker rlf c= rightker qf
  proof
    let x be object;
    assume x in rightker rlf;
    then consider ww be Vector of wqr such that
A7: x=ww and
A8: for vv be Vector of vq holds rlf.(vv,ww)=0.K;
    reconsider w = ww as Vector of wq by Th61;
    now
      let vv be Vector of vq;
      thus qf.(vv,w) = rlf.(vv,ww) by Th63
        .= 0.K by A8;
    end;
    hence thesis by A7;
  end;
  thus leftker qf c= leftker lrf
  proof
    let x be object;
    assume x in leftker qf;
    then consider vv be Vector of vq such that
A9: x=vv and
A10: for ww be Vector of wq holds qf.(vv,ww)=0.K;
    reconsider v=vv as Vector of vql by Th62;
    now
      let ww be Vector of wq;
      thus lrf.(v,ww) = qf.(vv,ww) by Th63
        .= 0.K by A10;
    end;
    hence thesis by A9;
  end;
  thus leftker lrf c= leftker qf
  proof
    let x be object;
    assume x in leftker lrf;
    then consider vv be Vector of vql such that
A11: x=vv and
A12: for ww be Vector of wq holds lrf.(vv,ww)=0.K;
    reconsider v=vv as Vector of vq by Th62;
    now
      let ww be Vector of wq;
      thus qf.(v,ww) = lrf.(vv,ww) by Th63
        .= 0.K by A12;
    end;
    hence thesis by A11;
  end;
  thus rightker qf c= rightker lrf
  proof
    let x be object;
    assume x in rightker qf;
    then consider ww be Vector of wq such that
A13: x=ww and
A14: for vv be Vector of vq holds qf.(vv,ww)=0.K;
    now
      let vv be Vector of vql;
      reconsider v = vv as Vector of vq by Th62;
      thus lrf.(vv,ww) = qf.(v,ww) by Th63
        .= 0.K by A14;
    end;
    hence thesis by A13;
  end;
  let x be object;
  assume x in rightker lrf;
  then consider ww be Vector of wq such that
A15: x=ww and
A16: for vv be Vector of vql holds lrf.(vv,ww)=0.K;
  now
    let vv be Vector of vq;
    reconsider v = vv as Vector of vql by Th62;
    thus qf.(vv,ww) = lrf.(v,ww) by Th63
      .= 0.K by A16;
  end;
  hence thesis by A15;
end;
