reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem Th93:
  x <> y & y <> z & z <> x iff <* x,y,z *> is one-to-one
proof
  set p = <* x,y,z *>;
A1: p.1 = x;
A2: p.3 = z;
  thus x <> y & y <> z & z <> x implies <* x,y,z *> is one-to-one
  proof
    assume that
A3: x <> y and
A4: y <> z and
A5: z <> x;
    {x,y} /\ {z} = {}
    proof
      set y1 = the Element of {x,y} /\ {z};
      assume
A6:   not thesis;
      then y1 in {x,y} by XBOOLE_0:def 4;
      then
A7:   y1 = x or y1 = y by TARSKI:def 2;
      y1 in {z} by A6,XBOOLE_0:def 4;
      hence thesis by A4,A5,A7,TARSKI:def 1;
    end;
    then {} = rng<* x,y *> /\ {z} by FINSEQ_2:127
      .= rng<* x,y *> /\ rng<* z *> by FINSEQ_1:38;
    then
A8: rng<* x,y *> misses rng<* z *>;
A9: <* z *> is one-to-one by Th91;
    <* x,y *> is one-to-one by A3,Th92;
    then <* x,y *> ^ <* z *> is one-to-one by A8,A9,Th89;
    hence thesis by FINSEQ_1:43;
  end;
A10: p.2 = y;
  1 in {1,2,3} by ENUMSET1:def 1;
  then
A11: 1 in dom p by Th1,FINSEQ_1:89;
  3 in {1,2,3} by ENUMSET1:def 1;
  then
A12: 3 in dom p by Th1,FINSEQ_1:89;
  2 in {1,2,3} by ENUMSET1:def 1;
  then
A13: 2 in dom p by Th1,FINSEQ_1:89;
  assume <* x,y,z *> is one-to-one;
  hence thesis by A11,A13,A12,A1,A10,A2;
end;
