let a, b be Complex; :: thesis: for X being ComplexUnitarySpace
for seq being sequence of X holds (a + b) * seq = (a * seq) + (b * seq)
let X be ComplexUnitarySpace; :: thesis: for seq being sequence of X holds (a + b) * seq = (a * seq) + (b * seq)
let seq be sequence of X; :: thesis: (a + b) * seq = (a * seq) + (b * seq)

now :: thesis: for n being Element of NAT holds ((a + b) * seq) . n = ((a * seq) + (b * seq)) . n

let n be Element of NAT ; :: thesis: ((a + b) * seq) . n = ((a * seq) + (b * seq)) . n
thus ((a + b) * seq) . n = (a + b) * (seq . n) by CLVECT_1:def 14
.= (a * (seq . n)) + (b * (seq . n)) by CLVECT_1:def 3
.= ((a * seq) . n) + (b * (seq . n)) by CLVECT_1:def 14
.= ((a * seq) . n) + ((b * seq) . n) by CLVECT_1:def 14
.= ((a * seq) + (b * seq)) . n by NORMSP_1:def 2 ; :: thesis: verum

end;

hence (a + b) * seq = (a * seq) + (b * seq) by FUNCT_2:63; :: thesis: verum