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Claim any two natural a,b are equal

By induction

Let m= max{a,b}

if m=1 then a=b=1 since a,b natural

suppose it is hold for m=k

if

max{a,b} = k then a=b

test if

max{a,b} = k+1 , sub 1

max{a-1,b-1} = k which is the previous so a-1 = b-1 , a=b

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