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 Th37:
  f is rotation implies
   (Det AutMt f = 1.F_Real iff f is base_rotation)
proof
  set TR=TOP-REAL n,cTR=the carrier of TR;
  set M=AutMt f;
  set MM=Mx2Tran M;
  A1: len M=n & width M=n by MATRIX_0:24;
  A2: n=0 implies n=0;
  A3: MM=f by Def6;
  assume A4: f is rotation;
  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 A3,Th36;
  set R=AutMt h;
  A7: width R=n by MATRIX_0:24;
  A8: h=Mx2Tran R by Def6;
  A9: AutMt(h*MM)=1.(F_Real,n) or AutMt(h*MM)=AxialSymmetry(n,n)
  by A4,A3,A5,A6,Th35;
  Det M=1.F_Real implies MM is base_rotation
  proof
    assume A10: Det M=1.F_Real;
    Det R=1.F_Real & n in NAT by A5,A8,Lm10,ORDINAL1:def 12;
    then
A11: Det(M*R)=(1.F_Real)*(1.F_Real) by A10,MATRIXR2:45
    .=1*1.F_Real
    .=1.F_Real
    .= 1;
    A12: rng MM c=cTR by RELAT_1:def 19;
    A13: rng h=[#]TR by A5,TOPS_2:def 5;
    A14: dom h=[#]TR & h is one-to-one by A5,TOPS_2:def 5;
    A15: dom(h")=[#]TR by A5,TOPS_2:def 5;
    A16: id TR=Mx2Tran 1.(F_Real,n) by MATRTOP1:33;
    h*MM=id cTR
    proof
      assume A17: h*MM<>id cTR;
      n<>0
      proof
        A18: dom(h*MM)=cTR & dom id cTR=cTR by FUNCT_2:52;
        assume  n=0;
        then A19: 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 A19,ZFMISC_1:33;
        hence contradiction by A17,A18,A19,FUNCT_1:7;
      end;
      then A20: n in Seg n by FINSEQ_1:3;
      Mx2Tran AutMt(h*MM)=h*MM by Def6;
      then Mx2Tran AxialSymmetry(n,n)=Mx2Tran(M*R)
        by A1,A2,A9,A8,A7,A16,A17,MATRTOP1:32;
      then AxialSymmetry(n,n)=M*R by MATRTOP1:34;
      then Det AxialSymmetry(n,n)= Det(M*R);
      hence contradiction by A11,A20,Th4;
    end;
    then h"*(h*MM) =h" by A15,RELAT_1:52;
    then h"=(h"*h)*MM by RELAT_1:36
    .=(id cTR)*MM by A14,A13,TOPS_2:52
    .=MM by A12,RELAT_1:53;
    hence thesis by A5;
  end;
  hence thesis by A3,Lm10;
end;
