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1.Rxh7+ Kxh7
2.Qh4+ Kg8 (2…Kg7,3.Qg5+ Kh7/8,4.Rh3#)
3.Rg3+ Qg7
4.Rxg7+ Kxg7
5.f6+ Kg8
6.Qg5+ Kh7/8
7.Qg7#
if black refuses to take the rook on h7 by
1.Rxh7+ Kg8
2.Qg3+ Kxh7
4.Qh4+ … and so on
the line is extended to a mate in 8 moves, isn’t it? or is there an other solution?
i think the only crucial think is finding 1, Rxh7+ on the 1st place and keep checking all the way since white cannot let any chance for Black to check or it will be 0-1.
Black can also play 5. … Kg6 but then 6. Qg5+ Kh7 7. Qg7#