let A be Interval; :: thesis: for x being Real st 0 <= x holds
for y being Real st y = diameter A holds
x * y = diameter (x ** A)
let x be Real; :: thesis: ( 0 <= x implies for y being Real st y = diameter A holds
x * y = diameter (x ** A) )
assume A1: 0 <= x ; :: thesis: for y being Real st y = diameter A holds
x * y = diameter (x ** A)
let y be Real; :: thesis: ( y = diameter A implies x * y = diameter (x ** A) )
assume A2: y = diameter A ; :: thesis: x * y = diameter (x ** A)

per cases ( A is empty or not A is empty ) ;

suppose A3: A is empty ; :: thesis: x * y = diameter (x ** A)

A4: now

assume not x ** A is empty ; :: thesis: contradiction
then consider y being real number such that
A5: y in x ** A by MEMBERED:9;
reconsider y = y as Real by XREAL_0:def 1;
ex z being Real st
( z in A & y = x * z ) by A5, INTEGRA2:39;
hence contradiction by A3; :: thesis: verum

end;

thus x * y = 0 by A2, A3, MEASURE5:10
.= diameter (x ** A) by A4, MEASURE5:10 ; :: thesis: verum

end;

suppose A6: not A is empty ; :: thesis: x * y = diameter (x ** A)

then consider z being real number such that
A7: z in A by MEMBERED:9;
reconsider z = z as Real by XREAL_0:def 1;
A8: x * z in x ** A by A7, MEMBER_1:193;
reconsider AA = A as non empty Subset of REAL by A6;
reconsider u = x as R_eal by XXREAL_0:def 1;
A9: inf (x ** AA) = u * (inf AA) by A1, Th21;
reconsider z = x as R_eal by XXREAL_0:def 1;
thus x * y = z * (diameter A) by A2, EXTREAL1:1
.= z * ((sup A) - (inf A)) by A6, MEASURE5:def 6
.= (z * (sup A)) - (z * (inf A)) by XXREAL_3:100
.= (sup (x ** A)) - (inf (x ** A)) by A1, A9, Th20
.= diameter (x ** A) by A8, MEASURE5:def 6 ; :: thesis: verum

end;

end;