reserve x,X for set,
        r,r1,r2,s for Real,
        i,j,k,m,n for Nat;
reserve p,q for Point of TOP-REAL n;
reserve f,f1,f2 for homogeneous additive Function of TOP-REAL n,TOP-REAL n;

theorem Th38:
  f is rotation implies
   (Det AutMt f = 1.F_Real or Det AutMt f = -1.F_Real)
proof
  set M=AutMt f;
  set MM=Mx2Tran M,TR=TOP-REAL n;
  A1: len M=n & width M=n by MATRIX_0:24;
  A2: n=0 implies n=0;
  n=0 or n>=1 by NAT_1:14;
  then A3: Det 1.(F_Real,n)=1_F_Real & n>=1 or Det 1.(F_Real,n)=1.F_Real & n=0
    by MATRIXR2:41,MATRIX_7:16;
  assume f is rotation;
  then A4: MM is rotation by Def6;
  then consider h be homogeneous additive Function of TR,TR such that
  A5: h is base_rotation and
  A6: h*MM is{n}-support-yielding by Th36;
  set R=AutMt h;
  A7: h=Mx2Tran R by Def6;
  then n in NAT & Det R=1.F_Real by A5,Lm10,ORDINAL1:def 12;
  then A8: Det(M*R)=Det M*(1.F_Real) by MATRIXR2:45
  .=Det M*1
  .=Det M;
  width R=n by MATRIX_0:24;
  then A9: h*MM=Mx2Tran(M*R) by A1,A2,A7,MATRTOP1:32;
  per cases by A4,A5,A6,Th35;
  suppose AutMt(h*MM)=1.(F_Real,n);
    hence thesis by A3,A8,A9,Def6;
  end;
  suppose A10: AutMt(h*MM)<>1.(F_Real,n) & AutMt(h*MM)=AxialSymmetry(n,n);
    set cTR=the carrier of TR;
    n<>0
    proof
      A11: dom(h*MM)=cTR & dom id cTR=cTR by FUNCT_2:52;
      assume  n=0;
      then A12: cTR ={0.TR} by EUCLID:22,77;
      rng(h*MM)c=cTR by RELAT_1:def 19;
      then rng id cTR=cTR & rng(h*MM)=cTR by A12,ZFMISC_1:33;
      then h*MM=id TR by A11,A12,FUNCT_1:7
      .=Mx2Tran 1.(F_Real,n) by MATRTOP1:33;
      hence contradiction by A10,Def6;
    end;
    then n in Seg n by FINSEQ_1:3;
    then Det AxialSymmetry(n,n)=-1.F_Real by Th4;
    hence thesis by A8,A9,A10,Def6;
  end;
end;
