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 Th92:
  x <> y iff <* x,y *> is one-to-one
proof
A1: <* x,y *>.2 = y;
  2 in {1,2} by TARSKI:def 2;
  then
A2: 2 in dom<* x,y *> by FINSEQ_1:2,89;
  thus x <> y implies <* x,y *> is one-to-one
  proof
    assume
A3: x <> y;
    let y1,y2 be object;
    assume that
A4: y1 in dom<* x,y *> and
A5: y2 in dom<* x,y *> and
A6: <* x,y *>.y1 = <* x,y *>.y2;
A7: y2 in {1,2} by A5,FINSEQ_1:2,89;
A8: y1 in {1,2} by A4,FINSEQ_1:2,89;
    now
      per cases by A8,A7,TARSKI:def 2;
      suppose
        y1 = 1 & y2 = 1 or y1 = 2 & y2 = 2;
        hence thesis;
      end;
      suppose
A9:     y1 = 1 & y2 = 2;
        thus thesis by A3,A6,A9;
      end;
      suppose
A10:    y1 = 2 & y2 = 1;
        thus thesis by A3,A6,A10;
      end;
    end;
    hence thesis;
  end;
  assume that
A11: <* x,y *> is one-to-one and
A12: x = y;
  1 in {1,2} by TARSKI:def 2;
  then
A13: 1 in dom<* x,y *> by FINSEQ_1:2,89;
  <* x,y *>.1 = x;
  hence thesis by A11,A12,A13,A2,A1;
end;
