Example 3 revisited



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H:= x^2+y^2-(x-e)^2-y^2=0, T:=y=0
x^2+y^2-(x-a)^2-(y-b)^2=0,


and we have checked that the implication H===>T is false.


So, we must try

EITHER

H=0 & (H=0 &T=0) ===> T=0


that is

x^2+y^2-(x-e)^2-y^2=0 and x^2+y^2-(x-a)^2-(y-b)^2=0

and
yt-1=0

which yields

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Elim(x..t,Ideal(x^2+y^2-(x-e)^2-y^2,
H=0
x^2+y^2-(x-a)^2-(y-b)^2,

yt-1)) (T=0)

Ideal(0)
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ie. H=0 & (0=0) ===> T=0

which is a trivial theorem.




OR


H=0 & (H=0& T=0) ===>T=0

that is

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Elim(x..t,Ideal(x^2+y^2-(x-e)^2-y^2,
H=0
x^2+y^2-(x-a)^2-(y-b)^2,

y)) T=0

Ideal(-1/2a^2e - 1/2b^2e + 1/2ae^2)


ie. H=0 & (-a^2e - b^2e + ae^2=0) ===> T=0


which is a more interesting theorem.


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