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 Th39:
  f1 is rotation & Det AutMt f1 = -1.F_Real & i in Seg n &
  AutMt f2 = AxialSymmetry(i,n) implies f1*f2 is base_rotation
 proof
   set M=AutMt f1;
   set A=AutMt f2;
   assume that
   A1: f1 is rotation and
   A2: Det M=-1.F_Real and
   A3: i in Seg n and
   A4: A=AxialSymmetry(i,n);
   A5: f2=Mx2Tran AxialSymmetry(i,n) by A4,Def6;
   reconsider MF=(Mx2Tran M)*f2 as homogeneous additive Function of TOP-REAL n,
   TOP-REAL n;
   set A=AxialSymmetry(i,n);
   set R=AutMt MF;
   A6: n=0 implies n=0;
   A7: MF=Mx2Tran R & width M=n by Def6,MATRIX_0:24;
   len A=n & width A=n by MATRIX_0:24;
   then Mx2Tran R=Mx2Tran(A*M) by A5,A6,A7,MATRTOP1:32;
   then A8: R=A*M by MATRTOP1:34;
   n in NAT & Det A=-1.F_Real by A3,Th4,ORDINAL1:def 12;
   then A9: Det R=(-1.F_Real)*(-1.F_Real) by A2,A8,MATRIXR2:45
   .=(-1.F_Real)*(-1)
   .=1.F_Real;
   A10: Mx2Tran M=f1 by Def6;
   f2 is rotation by A3,A5,Th27;
   hence thesis by A10,A1,A9,Th37;
 end;
