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The method by which mathematics arrives at its equations is the method of substitution.

For equations express the substitutability of two expressions, and we proceed from a number of equations to new equations, replacing expressions by others in accordance with the equations.

6.241    Thus the proof of the proposition  2 2 = 4  runs:

                           ( OMEGA v)'x= OMEGA 'x   Def.
   OMEGA 22'x = ( OMEGA 2)2'x = ( OMEGA 2)1+1'x =  OMEGA 2' OMEGA 2'x =  OMEGA 1+1' OMEGA 1+1'x
      = ( OMEGA ' OMEGA )'( OMEGA ' OMEGA )'x =  OMEGA ' OMEGA ' OMEGA ' OMEGA 'x =  OMEGA 1+1+1+1'x =  OMEGA 4'x.