Invent a formal system MULT that is similar to ADD but whose

Invent a formal system, MULT, that is similar to ADD but whose theorems are true statements about multiplication of positive integers. Then show that the formula II times III = IIIIII is a theorem of the system

Solution

The formal formula ||x|||=||||||is te theorem of the formal system MULT . To show this , it suffices
it should follow this formal proof. no of | of left of x and no of | on right of x multiplied is equal
to the no of | on right of =.

1) |X|=|
2) ||X|=||
3) |||X|=||||
4) |X|||=|||
5) ||X|||=||||||

given a formula system F and a property Q of formulas we often want to prove that evey theorm of F has
property Q .every axiom of F has the property Q, for each rule of inference of F if each hypothesis
of the rule has the property Q then the conclusion also has property Q.
S(1) states that a theorem with a proof in one step hasproperty Q where is basic property step of the formal system.
for N>1 assume that s(1),...s(n) are all true now find for n+1 step .let A have the proof in n+1 step so An+1=A.
by An+1 has property Q provided that each hypothesis has property Q. thus each hypothesis has the property Q .
hence An+1 has the property Q as required.

As no of | of left of x and no of | on right of x multiplied is equal
to the no of | on right of = so ||*|||=|||||| is true according to above proof.