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
  n = 1 & |.p.| = |.q.| implies
    ex f st f is rotation & f.p=q &
           (AutMt f = AxialSymmetry(n,n) or AutMt f = 1.(F_Real,n))
proof
  set TR=TOP-REAL n;
  assume that
  A1: n=1 and
  A2: |.p.|=|.q.|;
  per cases;
  suppose A3: p=q;
    take I=id TR;
    id TR=Mx2Tran 1.(F_Real,1) by A1,MATRTOP1:33;
    hence thesis by A1,A3,Def6;
  end;
  suppose A4: p<>q;
    A5: len p=1 by A1,CARD_1:def 7;
    then A6: p=<*p.1*> by FINSEQ_1:40;
    A7: 1 in Seg 1;
    then reconsider f=Mx2Tran AxialSymmetry(1,1) as
      homogeneous additive rotation Function of TR,TR by A1,Th27;
    take f;
    A8: (q.1)^2>=0 & (p.1)^2>=0 by XREAL_1:63;
    reconsider pk =(p.1)^2, qk = (q.1)^2 as Real;
    A9: |.p.|=sqrt Sum sqr<*p.1*> by A5,FINSEQ_1:40
    .=sqrt Sum<*pk*> by RVSUM_1:55
    .=sqrt((p.1)^2) by RVSUM_1:73;
    A10: len q=1 by A1,CARD_1:def 7;
    then A11: q=<*q.1*> by FINSEQ_1:40;
    |.q.|=sqrt Sum sqr<*q.1*> by A10,FINSEQ_1:40
    .=sqrt Sum<*qk*> by RVSUM_1:55
    .=sqrt((q.1)^2) by RVSUM_1:73;
    then A12: (q.1)^2=(p.1)^2 by A2,A8,A9,SQUARE_1:28;
    len(f.p)=1 by A1,CARD_1:def 7;
    then f.p=<*f.p.1*> by FINSEQ_1:40
    .=<*-p.1*> by A1,A7,Th9
    .=q by A4,A6,A11,A12,SQUARE_1:40;
    hence thesis by A1,Def6;
  end;
end;
