per cases ( n is zero or not n is zero ) ;

suppose n is zero ; :: thesis: for b₁ being Relation of (n -tuples_on X) st b₁ = n -placesOf R holds
( b₁ is transitive & b₁ is total )

hence for b₁ being Relation of (n -tuples_on X) st b₁ = n -placesOf R holds
( b₁ is transitive & b₁ is total ) by Lm8; :: thesis: verum

end;

suppose not n is zero ; :: thesis: for b₁ being Relation of (n -tuples_on X) st b₁ = n -placesOf R holds
( b₁ is transitive & b₁ is total )

hence for b₁ being Relation of (n -tuples_on X) st b₁ = n -placesOf R holds
( b₁ is transitive & b₁ is total ) by Lm7; :: thesis: verum

end;

end;