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;

theorem Th11:
  i in Seg n implies Mx2Tran AxialSymmetry(i,n) is {i}-support-yielding
proof
  set M=Mx2Tran AxialSymmetry(i,n);
  assume A1: i in Seg n;
  let f be Function,x be set;
  assume f in dom M;
  then reconsider F=f as Point of TOP-REAL n by FUNCT_2:52;
  assume A2: M.f.x<>f.x;
  len(M.F)=n by CARD_1:def 7;
  then A3: dom(M.F)=Seg n by FINSEQ_1:def 3;
  A4: len F=n by CARD_1:def 7;
  then A5: dom F=Seg n by FINSEQ_1:def 3;
  per cases;
  suppose A6: not x in Seg n;
    then M.F.x={} by A3,FUNCT_1:def 2
    .=F.x by A5,A6,FUNCT_1:def 2;
    hence thesis by A2;
  end;
  suppose x in Seg n;
    then x=i by A1,A2,A4,Th8;
    hence thesis by TARSKI:def 1;
  end;
end;
