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 Th34:
  i in Seg n & n >= 2 implies ex f st f is base_rotation & f.p = p+*(i,-p.i)
proof
  set TR=TOP-REAL n;
  assume that
  A1: i in Seg n and
  A2: n>=2;
  A3: {i}c=Seg n by A1,ZFMISC_1:31;
  A4: 1<=i by A1,FINSEQ_1:1;
  card Seg n=n & card{i}=1 by CARD_2:42,FINSEQ_1:57;
  then {i}<>Seg n by A2;
  then {i}c<Seg n by A3,XBOOLE_0:def 8;
  then consider j be object such that
  A5: j in Seg n and
  A6: not j in {i} by XBOOLE_0:6;
  reconsider j as Nat by A5;
  A7: j<>i by A6,TARSKI:def 1;
  A8: 1<=j by A5,FINSEQ_1:1;
  set p0=p+*(i,-p.i);
  A9: len p0=len p by FUNCT_7:97;
  A10: len p=n by CARD_1:def 7;
  then A11: dom p=Seg n by FINSEQ_1:def 3;
  A12: i<=n by A1,FINSEQ_1:1;
  (p.i)*(p.i)>=0 & (p.j)*(p.j)>=0 by XREAL_1:63;
  then A13: 0^2=0*0 & 0<=(p.i)^2+(p.j)^2;
  A14: j<=n by A5,FINSEQ_1:1;
  per cases by A7,XXREAL_0:1;
  suppose A15: i<j;
    then consider r such that
    A16: (Mx2Tran Rotation(i,j,n,r)).p.i=0 by A4,A13,A14,Th24;
    set s=sin r,c =cos r;
    A17: 0=p.i*c+p.j*(-s) by A4,A14,A15,A16,Th21;
    reconsider M=Mx2Tran Rotation(i,j,n,r+r) as
      base_rotation Function of TR,TR by A4,A14,A15,Th28;
    set Mp=M.p;
    A18: cos(r+r)=c*c-s*s & sin(r+r)=s*c+s*c by SIN_COS:75;
    A19: M is{i,j}-support-yielding by A4,A14,A15,Th26;
    A20: for k st 1<=k & k<=n holds p0.k=Mp.k
    proof
      let k;
      assume 1<=k & k<=n;
      then A21: k in Seg n;
      per cases;
      suppose A22: i=k;
        hence p0.k= -p.i*1 by A11,A21,FUNCT_7:31
        .=-p.i*(s*s+c*c) by SIN_COS:29
        .=p.i*(c*c-s*s)-(p.i*c*c)-(p.i*c*c)
        .=p.i*(c*c-s*s)-(p.j*s*c)-(p.j*s*c) by A17
        .=p.i*(c*c-s*s)+p.j*(-(s*c+s*c))
        .=Mp.k by A4,A14,A15,A18,A22,Th21;
      end;
      suppose A23: j=k;
        hence p0.k=p.j*1 by A15,FUNCT_7:32
        .=p.j*(s*s+c*c) by SIN_COS:29
        .=p.j*s*s+p.j*s*s+p.j*(c*c-s*s)
        .=(p.i*c*s)+(p.i*c*s)+p.j*(c*c-s*s) by A17
        .=p.i*(s*c+s*c)+p.j*(c*c-s*s)
        .=Mp.k by A4,A14,A15,A18,A23,Th22;
      end;
      suppose A24: i<>k & j<>k;
        A25: dom M=the carrier of TR by FUNCT_2:52;
        p0.k=p.k & not k in {i,j} by A24,FUNCT_7:32,TARSKI:def 2;
        hence thesis by A19,A25;
      end;
    end;
    take M;
    len Mp=n by CARD_1:def 7;
    hence thesis by A9,A10,A20;
  end;
  suppose A26: j<i;
    then consider r such that
    A27: (Mx2Tran Rotation(j,i,n,r)).p.i=0 by A8,A12,A13,Th25;
    set s=sin r,c =cos r;
    A28: 0=p.j*s+p.i*c by A8,A12,A26,A27,Th22;
    reconsider M=Mx2Tran Rotation(j,i,n,r+r) as
      base_rotation Function of TR,TR by A8,A12,A26,Th28;
    set Mp=M.p;
    A29: cos(r+r)=c*c-s*s & sin(r+r)=s*c+s*c by SIN_COS:75;
    A30: M is{i,j}-support-yielding by A8,A12,A26,Th26;
    A31: for k st 1<=k & k<=n holds p0.k=Mp.k
    proof
      let k;
      assume 1<=k & k<=n;
      then A32: k in Seg n;
      per cases;
      suppose A33: i=k;
        hence p0.k  =-p.i*1 by A11,A32,FUNCT_7:31
        .=-p.i*(s*s+c*c) by SIN_COS:29
        .=(-p.i*c*c)+(-p.i*c)*c+p.i*(c*c-s*s)
        .=p.j*s*c+p.j*s*c+p.i*(c*c-s*s) by A28
        .=p.j*(s*c+s*c)+p.i*(c*c-s*s)
        .=Mp.k by A8,A12,A26,A29,A33,Th22;
      end;
      suppose A34: j=k;
        hence p0.k=p.j*1 by A26,FUNCT_7:32
        .=p.j*(s*s+c*c) by SIN_COS:29
        .=p.j*(c*c-s*s)+(p.j*s)*s+(p.j*s)*s
        .=p.j*(c*c-s*s)+(-p.i*c)*s+(-p.i*c)*s by A28
        .=p.j*(c*c-s*s)+p.i*(-(s*c+s*c))
        .=Mp.k by A8,A12,A26,A29,A34,Th21;
      end;
      suppose A35: i<>k & j<>k;
        A36: dom M=the carrier of TR by FUNCT_2:52;
        p0.k=p.k & not k in {i,j} by A35,FUNCT_7:32,TARSKI:def 2;
        hence thesis by A30,A36;
      end;
    end;
    take M;
    len Mp=n by CARD_1:def 7;
    hence thesis by A9,A10,A31;
  end;
end;
