reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;
reserve i,j,k for Element of omega;
reserve x,y,z for Element of RAT+;
reserve i,j,k for natural Ordinal;

theorem Th52:
  (x*'y)*'z = x*'(y*'z)
proof
  set nx = numerator x, ny = numerator y, nz = numerator z;
  set dx = denominator x, dy = denominator y, dz = denominator z;
A1: x = nx/dx by Th39;
  z = nz/dz by Th39;
  hence (x*'y)*'z = (nx*^ny*^nz)/(dx*^dy*^dz) by Th49
    .= (nx*^(ny*^nz))/(dx*^dy*^dz) by ORDINAL3:50
    .= (nx*^(ny*^nz))/(dx*^(dy*^dz)) by ORDINAL3:50
    .= x*'(y*'z) by A1,Th49;
end;
