reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;
reserve z for Complex;

theorem
  for x,y,z,t being positive Nat holds x/y + y/z + z/t + t/x <> 1
  proof
    let x,y,z,t be positive Nat;
    x/y >= 1 or y/z >= 1 or z/t >= 1 or t/x >= 1
    proof
      assume that
A1:   x/y < 1 & y/z < 1 and
A2:   z/t < 1 and
A3:   t/x < 1;
A4:   (x/y) * (y/z) = (x*y)/(y*z) by XCMPLX_1:76;
A5:   ((x*y*z) / (y*z*t)) * (t/x) = (x*y*z*t) / (y*z*t*x) by XCMPLX_1:76
      .= 1 by XCMPLX_1:60;
A6:   (x/y) * (y/z) * (z/t) = (x*y*z)/(y*z*t) by A4,XCMPLX_1:76;
      (x/y) * (y/z) < 1 by A1,Th54;
      then (x/y) * (y/z) * (z/t) < 1 by A2,A4,Th54;
      hence contradiction by A3,A5,A6,Th54;
    end;
    then per cases;
    suppose x/y >= 1;
      then x/y + (y/z + z/t + t/x) > 1+0 by XREAL_1:8;
      hence thesis;
    end;
    suppose y/z >= 1;
      then y/z + (x/y + z/t + t/x) > 1+0 by XREAL_1:8;
      hence thesis;
    end;
    suppose z/t >= 1;
      then z/t + (x/y + y/z + t/x) > 1+0 by XREAL_1:8;
      hence thesis;
    end;
    suppose t/x >= 1;
      then t/x + (x/y + y/z + z/t) > 1+0 by XREAL_1:8;
      hence thesis;
    end;
  end;
